Uniform Acceleration
Forces and Motion

Acceleration due to gravity

for 14-16

Gravity plays a role in the motion of many common objects. These experiments confirm that the vertical component of motion is properly described as an acceleration. And they yield a value worth memorizing.

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Investigating free fall with a light gate

Uniform Acceleration
Forces and Motion

Investigating free fall with a light gate

Practical Activity for 14-16

Demonstration

The acceleration of an object allowed to fall under the force of gravity is found by dropping a card vertically through a light gate. The emphasis of this datalogging experiment is on investigating the relationship between the velocity of the card and the distance it has fallen from rest.

Apparatus and Materials

  • Light gate, interface and computer
  • Weighted card
  • Clamp and stand
  • Metre or half metre rule

Health & Safety and Technical Notes

Read our standard health & safety guidance


Clamp the light gate about 20 cm above the bench. Clamp a ruler so that the vertical distance may be measured from above the level of the light gate.

Cut black card to the precise length of 10.0 cm. Draw a pencil line across the width of the card at exactly half its length.

Measurements of the height fallen by the card should be made to this line rather than the lower or upper edge of the card. Adding two small blobs of Blutack, at the lower corners, will improve the stability of the card as it falls.

Configure the data-logging software to measure the transit time and calculate the velocity as the card passes through the light gate. A series of results is accumulated in a table. This should also include a column for the manual entry of distance measurements taken from the ruler.

Procedure

Data collection

  1. Hold the card above the light gate and next to the ruler so that its height above the gate may be measured carefully. Release the card so that it cuts through the light beam; a velocity measurement should appear in the table on the screen. Repeat this measurement from the same height several times; enter the height value in the height column of the table in the computer program.
  2. Repeat this procedure for a new starting height 2 cm above the first.
  3. Collect a series of measurements, each time increasing the height by 2 cm.

Analysis

  1. Depending upon the software, the results may be displayed on a bar chart as the experiment proceeds. Note the relative increase in values of velocity as greater heights are chosen.
  2. The relationship between velocity and height fallen is more precisely investigated by plotting a XY graph of these two quantities. (Y axis: velocity; X axis: height fallen.) Use a curve matching tool to identify the algebraic form of the relationship. This is usually of the form 'velocity is proportional to the square root of height'.
  3. Use the program to calculate a new column of data representing the square of the velocity. Plot this against height on a new graph. A straight line is the usual result, showing that the velocity squared is proportional to the height fallen.

Teaching Notes

  • One of the chief values of real-time data-logging, exemplified here, is the interaction between the collection and simultaneous display of results. These can be used to prompt students' thinking. You can orchestrate discussion as the experiment proceeds. If you do this as a class experiment, students should be prompted to ask questions about the results and the process involved in collecting them.
  • Small variations between individual results should be observed and possible sources of error discussed. For example, how clean was the release of the card, how precise was the height measurement, was the fall wobble-free? and so on. The visual display of results on a bar chart helps to show the significance of these variations, compared with overall trends in the relationship between velocity and height. The chart also makes it very easy to spot anomalous results due to the card wobbling or snagging on the ruler as it falls. The software usually facilitates the deletion or hiding of such values.
  • Software provides several alternative approaches to analysis. Two examples are described here; a straightforward plot of the collected data, velocity against height, and a second plot involving calculated data.
  • The first graph yields a curve which may be conveniently evaluated using a software curve-fitting tool. (It is best to give students tools which allow them to experiment with simple general forms such as quadratics and power laws rather than polynomial fits whose physical significance is hard to interpret.)
  • The second graph yields a straight line which lends itself easily to conventional analysis using concepts such as gradient and intercept.
  • Note: The discussion here has assumed that the software is capable of calculating and showing velocity as the primary measurement. Physically the system actually measures transit times for the card passing through the light gate. It is necessary for the user to enter into the program the length of the card (the distance travelled during the transit time) at the beginning of the session. Calculations may then be performed as a matter of routine during the experiment. Eliminating this calculation step from the experimental procedure allows thinking and discussion to focus on the relationship between velocity and height fallen.

This experiment was safety-tested in May 2006

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Multiflash photographs of free fall

Uniform Acceleration
Forces and Motion

Multiflash photographs of free fall

Practical Activity for 14-16

Demonstration

This provides an interesting and graphic way of obtaining a value for the acceleration of a falling object. Click here for a video showing free fall from which you can make measurements (please note this runs only in Internet Explorer 4+).

Apparatus and Materials

  • Bail or other object to be photographed
  • Camera and stroboscope or multiflash system
  • Scale, 10 cm, with strong contrast
  • Lamp, bright (500 W) or slide projector
  • Box lined with soft material

Health & Safety and Technical Notes

Do not allow anyone to climb on stools or benches to release the ball. Use a pair of steps (stepladder) or a kick-stool.

Read our standard health & safety guidance


A steel ball (15 mm to 25 mm), golf ball, or lamp with cell are all suitable objects to drop.

Read the multiflash photography guidance page for detail of specific methods and for general hints:

Multiflash Photography


Procedure

The setup:

  1. Set up the arrangement as shown in the diagram.
  2. Darken the room and shine light on the falling object but not the background. See guidance page:

    Classroom mangement in semi-darkness


  3. Set the stroboscope rotating.
  4. Making an image:
  5. Prepare to drop the ball from near the top of the camera's field of view.
  6. Open the camera shutter using 3-2-1-0 countdown and drop the ball.
  7. Close the shutter when the ball lands or leaves the field of view.
  8. Record the image frequency in hertz, which is the same as the number of images per second. Use this to work out the time between each image, using time = 1/frequency. Frequency is measured in hertz (or images per second).
  9. Analysing the image:
  10. Choose the first image of the ball after its release. You can define 'time zero' to be the time at which this image was produced. Use the grid to measure the distance travelled by the object for each later image.
  11. Use the time and distance data to plot a distance-time graph, manually or with a computer. Draw a smooth curve to fit the points. It will not pass through the origin if t does not = 0 when s=0.
  12. The gradient of the distance-time graph, at any time, is equal in value to the velocity. Measure the gradient of the curve at several times, to obtain a set of values of time and velocity. Use this information to produce a velocity-time graph.
  13. The value of the gradient of the velocity-time graph is equal to the acceleration at any time. If the ball fell under gravity and if air resistance was not significant, then the velocity-time graph should be a straight line. The ball had a constant acceleration. This is the acceleration due to gravity.

Teaching Notes

  • The accepted value of the acceleration due to gravity is 9.81 m/s 2 . Any experimental value in the region of 10 m/s 2 is a reasonable one.
  • If air resistance is significant compared with the weight of the falling object, then the gradient of the speed-time graph will decrease. This indicates a decreasing acceleration caused by the increasing effect of air resistance matching the increasing speed of the object.
  • As an extension activity, it is worth comparing the fall of a golf ball with that of a polystyrene ball. In the latter case, air resistance is more significant relative to weight, and acceleration quickly decreases. Extension: investigate the effect of object mass on its acceleration. (Provided that air resistance is not significant, object mass should make no difference.)

This experiment was safety-tested in April 2006

Resources

Download the support sheet / student worksheet for this practical.

Up next

Motion of a ball rolling down a plank

Uniform Acceleration
Forces and Motion

Motion of a ball rolling down a plank

Practical Activity for 14-16

Demonstration and Class practical

This is a version of an experiment devised and carried out by Galileo. He used it to discover and demonstrate that motion was subject to quite simple, and beautiful, mathematical description.

Apparatus and Materials

  • Plank, long, with one grooved edge
  • Marbles, large, a supply of
  • Hanging pennants, metal or light plastic, up to 6
  • Clamps and stands for holding each pennant
  • String
  • Masses, 50 g, 5 or 6

Health & Safety and Technical Notes

The long plank should be handled by two persons.

Do not stand on a stool or bench to drop an object on a string. It could be dropped down a stairwell as long as precautions are taken to prevent the object falling on anyone.

Read our standard health & safety guidance


A length of 3 m of grooved plank is suitable. Alternatively, you could use a plain plank with grooved moulding fixed to it.

The pennants can be small rectangles of metal sheet, just large enough to make a clear clink as a marble hits one and rolls past. Christmas tree or similar decorations can add a touch of colour. They should make a clear sound without possibility of damage and without resisting the motion of the marble too much.

Horizontal rods held in clamps can be used to hang the pennants. Alternatively, you can fix wires from which to hang the pennants to curtain track fitted to the plank. Or you could make small wire goalposts which could sit over the grooved track.

It must be possible to move the pennant holders to different positions on the track. And whichever system you use, the pennants should be hung before the lesson.

Procedure

  1. Set up the grooved plank inclined so that marbles can roll down the groove.
  2. Hang small tin pennants above the groove so that the marbles hit them and make' clinks'.
  3. Place the pennants at regular intervals: 25 cm, 50 cm, 75 cm, 100 cm from the beginning of the plank. Roll a marble down the track, and listen for the time intervals between each clink.
  4. Try placing the pennants so that the clinks happen at equal time intervals.

Teaching Notes

  • Galileo showed that the total distance travelled increases with the square of the time. The consequence here is that, for the desired regular clink of the pennants, they will be at distances from the start which have the ratio 1:4:9:16 and so on.
  • As an extension activity, students could make graphs. They should first use values of time as 1, 2, 3, 4 units, plotted on the x -axis. They should plot the values of distance on the y -axis. They then make a second graph plotting the square of time on the x -axis, to obtain a straight line (unless friction plays a significant complicating role).
  • The same ratio, 1:4:9:16, occurs for all accelerations and hence, provided that frictional effects are not too great, for all slopes of the runway. Students could try this. Where graphs of distance against the square of the time are plotted, lines obtained for different runway slopes will have different gradients.
  • You can do a similar activity for a body in free fall, whose acceleration is the acceleration due to gravity. Tie a length of string to a mass. Tie knots in the string. The first one should be 10 centimetres from where the string is tied to the mass. The next one should be 40 centimetres away, the third one should be 90 centimetres away, and the fourth one 160 centimetres away. Let the string fall through your fingers from 160 cm above the floor with the mass falling first.
  • Feeling the knots is not as easy to discern as hearing masses hit the floor. Attach masses to the end of a string and at 10, 40, 90, 160, 250 cms from the original mass.
  • Hold the mass at the 250 cm position and lower the string out of a window or down a stairwell until the end mass reaches the floor. Leave go of the string, and the masses will hit the floor at equal time intervals.

This experiment was safety-tested in January 2005

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Multiflash photographs of motion down a slope

Uniform Acceleration
Forces and Motion

Multiflash photographs of motion down a slope

Practical Activity for 14-16

Demonstration

A steel ball rolls down a slope, starting from rest.

Apparatus and Materials

  • Steel ball, 15-25 mm in diameter, polished
  • Runway that will allow the ball to roll all the way down
  • Camera and multiflash system
  • Grid
  • Bright lamp, 500 W
  • Matt black background

Health & Safety and Technical Notes

Provide a catcher (person or box) to prevent the ball falling on the floor.

Read our standard health & safety guidance


Read the Multiflash Photography guidance page for general hints and detail of specific methods:

Multiflash Photography


A high frequency of exposure is required. This reduces any error in identifying the time of the start of the motion, relative to later images.

There should be a good contrast between the bright ball and its surroundings.

Procedure

Making the image

  1. Set up the runway with a slope. Place a grid behind it so that you are able to measure positions on the photographs. Darken the room. See guidance note:
  2. Classroom mangement in semi-darkness


  3. Start the multiflash system and then release the ball. Stop it when the ball passes out of the field of view.
  4. Analyzing the image
  5. Measure the distances between the positions of the ball. Use the known multiflash frequency to find the time interval between each position (time = 1/frequency).
  6. Plot two graphs: distance against time, and distance against time2.
  7. Compare the shapes of the graphs.
  8. The gradient of the distance - time graph is equal to velocity. Describe how the velocity changes.
  9. The gradient of the distance - time2 graph provides an average value for the acceleration of the ball. Work out the average acceleration.

Teaching Notes

  • Acceleration can be studied carefully using this variation on a freely falling object. Galileo is credited with the idea and performed the experiment with a rolling ball in 1604, publishing it in his Discourses Concerning Two New Sciences. He wanted to study the motion of a falling object, which moved too fast for his crude timing devices (his pulse or, in this case, a water clock). He came up with the idea of diluting the effect of gravity by only using a component of the full gravitation along the track. The beauty of this experiment lies in the dramatic and convincing way in which a relatively simple piece of equipment reveals the nature of acceleration due to gravity.
  • Measure the positions of the ball and look for a pattern. Most students will find Galileo's answer: the successive distances grow by jumps (3, 5, 7, 9 etc). However, the total distances themselves run as squares of the integers (1, 4, 9, 16 etc). (Measure the length of the first distance and see how many times longer the second distance is.) The start of the motion may not be clear on the image. The flash of light into the camera may not occur as the ball starts to move. Start the analysis where positions are distinct. The kind of results you might expect are shown in a results sheet (see below).
  • Some students might like to try to calculate the acceleration of the ball down the plank algebraically using the formula given below. You can calculate the average velocity between two consecutive positions of the ball near to the beginning of the run from:
    • The distance between the positions
    • The time taken to travel that distance.

Do the same for two positions of the ball near to the end of its run.

The acceleration is the change in velocity divided by the time taken for the change.

The formula to use is: a = v-ut

where a is the acceleration v is the second average velocity u is the first average velocity t is the time between the two average velocities.

Time t is the time between the midpoints of the two pairs of images used to find the average velocities (if the time represented by each image pair is not great). This is one less time interval than the number of intervals between the first and last ball images you have used.

This experiment was safety-tested in March 2005

Resources

Download the support sheet / student worksheet for this practical.

Up next

Building a reaction tester

Uniform Acceleration
Forces and Motion

Building a reaction tester

Practical Activity for 14-16

Class practical

To get the class to build a simple reaction tester using a falling sheet of card.

Apparatus and Materials

For each student pair

  • A3 card cut into strips
  • Metre rule

Health & Safety and Technical Notes

Read our standard health & safety guidance


The A3 card should be cut into strips, about 4 cm wide, the length of the short side of the paper.

Alternatively, plastic 30 cm rules can be used instead.

Procedure

  1. The students prepare a reaction timer, a sheet of card.
  2. The student whose reaction time is being tested places their thumb and first finger around the zero mark (without touching the card), while a partner holds the card vertically.
  3. When the partner releases the card, the first student closes the finger and thumb to catch it. The distance the card falls before being stopped is used to measure the reaction time.

Teaching Notes

  • At the simplest level a chart of distances can be used and ranked. At a higher level the teacher could supply a set of distances to be marked on the card for times of 0.1 s, 0.2 s, etc beginning with zero at the lowest level.
  • At a more advanced level still the students individually or as a whole class can be set the problem of putting on lines at times corresponding to the card falling for 0.1 s, 0.2 s, etc. They should use g = 10 m/s 2 for the calculations.
  • As students find most reaction times are in the area of 0.1 to 0.2 s they could consider marking lines more densely there. This leads into ideas about how to take data in non-uniform distributions. Students might also be led into the idea of taking averages and class surveys.
  • This links in with reaction time in stopping cars and is good practice with equations involving uniform acceleration. Distance versus time can be found by calculation using s=ut+1/2 at 2, u=o therefore s=5t 2or by graphs, with either produced by the student, by computer or by the teacher.

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Measurement of g using an electronic timer

Uniform Acceleration
Forces and Motion

Measurement of g using an electronic timer

Practical Activity for 14-16

Class practical

This experiment gives a direct measurement of the acceleration due to gravity.

Apparatus and Materials

  • Release mechanism (may be electromagnetic)
  • Trip switch (hinged flap)
  • Power supply, low voltage, DC
  • Switch, SPDT
  • Ball bearing ball, steel
  • Retort stand and boss
  • Electronic timer
  • Leads, 4 mm

Health & Safety and Technical Notes

It may be useful to stand the trip switch in an up-turned box lid to catch the ball after its fall.

Read our standard health & safety guidance


Science equipment suppliers offer slightly different versions of the mechanisms for the release that starts the timer and the trip switch that stops the timer.

i) In one version, there is no electromagnet in the release mechanism. Instead, the ball is held in position manually, so that it completes a circuit between two of three pegs. When you release the ball, the break in circuit starts the timer.

If you are using this arrangement, you may prefer to release the ball by holding it from a thread passing through the top of the support. This ensures that you do not obstruct the motion of the ball with your hand: the ball will be in free fall immediately the circuit is broken.

At the bottom of its fall, depending on its design, the mechanism may either make or break a second circuit, stopping the timer.

ii) Another version employs an electromagnet.

Procedure

  1. Set up the apparatus as shown in the diagram. You may need to adjust the distance of fall and the point at which the ball strikes the flap.
  2. Arrange the timer so that it starts when the electromagnet is switched off and stops when the hinged flap opens.
  3. Check that the flap does open when the ball strikes it. You may need to make the distance of fall larger, or move the flap so that the ball strikes it further from the hinge.
  4. Measure the distance h from the bottom of the ball to the hinged flap. Be careful to avoid parallax error in this measurement.
  5. Measure the fall time three times and find the average.
  6. Repeat step 5 for a range of heights between 0.5 m and 2.0 m.
  7. Plot a graph of 2h against t 2.
  8. Use the graph to find g.

Teaching Notes

  • The value of g is calculated from s=12at 2. In this case h=12gt 2.
  • With students at intermediate level, it will be sufficient to obtain an average value for t at just one height, h. Omit steps 7 and 8.
  • Steps 7 and 8: With more advanced students, repeat the experiment at different heights and find the gradient of the graph. They will see that the graph does not pass through the origin. An intercept on the t 2 axis indicates that there is an apparent time of fall even when the ball falls no distance at all, a systematic error. This is the time that it takes the electromagnet to release the ball once the current through it is switched off, in other words, the time for the magnetism to fall sufficiently to release the ball.
  • How Science Works Extension: This experiment provides an opportunity to discuss experimental design and how it can be used to reduce or eliminate errors. The experiment contains two sources of systematic error: the time-delay in releasing the ball (as discussed above in note 2), and a similar delay in switching the timer off, because the hinged flap may not switch the timer at exactly the instant when the ball strikes it. Both of these will give rise to measured times which are longer than the time during which the ball is in free-fall.
  • You could discuss with the class how to reduce or eliminate these errors. One approach is to improve the basic design of the experiment so that the time-delays are less or zero. You could compare this method of measuring g with others and discuss their relative merits.
  • Another approach is to consider how this experiment can be analyzed to reduce the systematic errors. The ‘true time of fall’ is less than the measured value t by a fixed amount terror (equal to the sum of the two time delays discussed above). How can we discover or otherwise eliminate this error? Here are two ways:
  • Think about the situation where h=0. The ball will take zero time to fall through this height, but the timer will still show a time equal to terror. Plot a graph of t against h. Extrapolate back to h=0. The graph will reach the axis at t=terror. Now this value can be used to correct the measured values of t. (Students will need extra columns in their results tables to allow for this.) Plot a graph of 2h against t 2 as before.
  • Alternatively, think about the situation where h is infinite, or at least very large). The value of t will be very large, so the error in it will be small or negligible. So large values of h will give more accurate values of g. In the results table, add a column for 2ht 2; the values should approach an accurate value for g. Plot a graph of 2h against t 2 as before; it will be curved, but at the high end it will approach a straight line through the origin whose gradient will be a good value of g.

This experiment was safety-checked in May 2005

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Multiflash photography

Conservation of Energy
Energy and Thermal Physics | Forces and Motion

Multiflash photography

Teaching Guidance for 14-16

Multiflash photography creates successive images at regular time intervals on a single frame.

Method 1: Using a digital camera in multiflash mode

You can transfer the image produced direct to a computer.

Method 2: Using a video camera

Play back the video frame by frame and place a transparent acetate sheet over the TV screen to record object positions.

Method 3: Using a camera and motor-driven disc stroboscope

You need a camera that will focus on images for objects as near as 1 metre away. The camera will need a B setting, which holds the shutter open, for continuous exposure. Use a large aperture setting, such as f3.5. Digital cameras provide an immediate image for analysis. With some cameras it may be necessary to cover the photocell to keep the shutter open.

Set up the stroboscope in front of the camera so that slits in the disc allow light from the object to reach the camera lens at regular intervals as the disc rotates.

Lens to disc distance could be as little as 1 cm. The slotted disc should be motor-driven, using a synchronous motor, so that the time intervals between exposures are constant.

You can vary the frequency of ‘exposure’ by covering unwanted slits with black tape. Do this symmetrically. For example, a disc with 2 slits open running at 300 rpm gives 10 exposures per second.

The narrower the slit, the sharper but dimmer the image. Strongly illuminating the objects, or using a light source as the moving object, allows a narrower slit to be used.

Illuminate the object as brightly as possible, but the matt black background as little as possible. A slide projector is a good light source for this purpose.

Method 4: Using a xenon stroboscope

This provides sharper pictures than with a disc stroboscope, provided that you have a good blackout. General guidance is as for Method 3. Direct the light from the stroboscope along the pathway of the object.

In multiflash photography, avoid flash frequencies in the range 15-20 Hz, and avoid red flickering light. Some people can feel unwell as a result of the flicker. Rarely, some people have photosensitive epilepsy.

General hints for success

You need to arrange partial blackout. See guidance note

Classroom management in semi-darkness


Use a white or silver object, such as a large, highly polished steel ball or a golf ball, against a dark background. Alternatively, use a moving source of light such as a lamp fixed to a cell, with suitable electrical connections. In this case, place cushioning on the floor to prevent breakage.

Use the viewfinder to check that the object is in focus throughout its motion, and that a sufficient range of its motion is within the camera’s field of view.

Place a measured grid in the background to allow measurement. A black card with strips of white insulating tape at, say, 10 cm spacing provides strong contrast and allows the illuminated moving object to stand out.

As an alternative to the grid, you can use a metre rule. Its scale will not usually be visible on the final image, but you can project a photograph onto a screen. Move the projector until the metre rule in the image is the same size as a metre rule held alongside the screen. You can then make measurements directly from the screen.

Use a tripod and/or a system of clamps and stands to hold the equipment. Make sure that any system is as rigid and stable as possible.

Teamwork matters, especially in Method 3. One person could control the camera, another the stroboscope system as necessary, and a third the object to be photographed.

  • Switch on lamp and darken room.
  • Check camera focus, f 3.5, B setting.
  • Check field of view to ensure that whole experiment will be recorded.
  • Line up stroboscope.
  • Count down 3-2-1-0. Open shutter just before experiment starts and close it as experiment ends.

Up next

Classroom management in semi-darkness

Interference
Light Sound and Waves

Classroom management in semi-darkness

Teaching Guidance for 14-16

There are some experiments which must be done in semi-darkness, for example, optics experiments and ripple tanks. You need to plan carefully for such lessons. Ensure that students are clear about what they need to do during such activities and they are not given unnecessary time. Keep an eye on what is going on in the class, and act quickly to dampen down any inappropriate behaviour before it gets out of hand.

Shadows on the ceiling will reveal movements that are not in your direct line of sight.

Up next

Applications of approximate methods of calculation

Uniform Acceleration
Forces and Motion

Applications of approximate methods of calculation

Teaching Guidance for 14-16

Approximate methods are widely used in pure science. Complex X-ray analysis (of proteins, for example) uses computers to produce maps of the molecules, whose shape is fitted by no simple equation. Approximate methods are used for predicting the structure of all atoms or molecules other than the very simplest, as the equations (Schrodinger's equation) cannot be solved exactly by analytic methods for more than two particles. No analytic equations have been found for astronomical problems involving several comparably sized bodies, and approximate numerical solutions rather than exact equations are used to guide space probes (and very exactly too).

On the other hand, approximate analytic solutions are used to guide numerical methods. If they were not, purely numerical methods would often defeat the largest computers imaginable.

In applied science, the role of approximate numerical methods is even wider. The airflow over an aircraft, the distribution of energy stored thermally in a building, the stresses in a proposed dam or bridge, the magnetic field around a new design of motor armature, or the effect of changing the shape of the hull of a ship are all examples where such methods, nowadays using computers, are the only possible approach in practical problems.

Engineers have pioneered new kinds of numerical calculation, ahead of the mathematicians. They are used for calculations on airflow, vibrations, transfer of energy in solids, deflection of spars of varying thickness, bridges, electrical networks, stability, torsion in beams or shafts, lubrication, structural frameworks, resonances (of aero engines and aircraft wings), stresses in hooks, shock waves, magnetic fields, flow through pipes and nozzles, and many other problems.

The availability of computers, which can quickly do simple repeated calculations, has made it far more practicable to tackle tough problems by approximate methods. Of course, the exact methods of differential and integral calculus are still valued, though more for their elegance, generality, and compactness than for their precision, because approximate methods can be made as exact as one pleases if one takes enough trouble.

Since computers are so much used for this kind of problem, there is merit in handling some problems in physics by teaching in a computational manner rather than an analytic one. Students who meet such methods later on may find them less strange than those who meet only analytic methods. But the main reason for using these methods at school level is that they can offer a better insight into both the meaning of derivatives and the way mathematics models a physical situation than do more formal methods. That is, they are more mathematical than is analysis.

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An approximate solution for constant acceleration

Uniform Acceleration
Forces and Motion

An approximate solution for constant acceleration

Teaching Guidance for 14-16

The following is suggested as a possible first example, for students, of numerical integration.

The phenomenon of a falling ball, accelerating at close to 10 m s-2, can quickly be given a graphical solution. Knowing that the distance fallen is given by s = ½at 2 also makes it possible to check the graphical technique.

Suppose the ball starts at rest, at time t = 0. Over a short time, say 0.1 second, around t = 0 the velocity is pretty well zero, the distance travelled is also nearly zero, and the first bit of the graph must be flat, like the segment AB in the figure above.

But in the following interval of another 0.1 second, around the 'time-of-day' t = 0.1 second the average velocity will equal that at that time. If the acceleration is 10 metres per second each second, the velocity is 1.0 metre per second. So the next segment of the graph, BC, in the same figure, rises or slopes up at 1.0 metre per second, rising in all 0.1 metre in an interval of 0.1 second.

The next 0.1 second interval centres around the time 0.2 second from the start, when the velocity will be averaging 10 x 0.2 = 2 metres per second. So the next section of the graph slopes up twice as steeply, rising 0.2 metre over 0.1 s econd. And so it goes on. The rule is easy: around each time t draw a section of line for velocity 10 t which must rise an extra distance (0.1) 10 t metre.

Note that for successive equal time intervals Δt, the velocity rises by the same amount each time. The acceleration is constant.

Such a graph is shown above (figure 2). The circles mark points calculated from s = ½at 2, for comparison. It predicts, for instance, that a heavy ball will fall 1 metre in 0.45 second. This could be tested using a scaler to time the fall, if your class feels the need. Or a ball could simply be dropped, and the class be asked to estimate the time. The equation s = ½at 2} and the kinked curve are both solutions of the equation:

acceleration = 10 m s-2 or, better, of = a where a = 10 m s-2.

The graph is an QuoteThis{approximate solution: it is near to the exact solution, and although where it is wrong it always makes the distance come out a little too small, the graph does not drift off course (that is, the errors do not accumulate). The smaller the interval Δt, the better the approximation. When solving real problems in this style, engineers and physicists devote much attention to choosing Δt small enough to be just sufficiently accurate, but not so small as to make the job unnecessarily tedious. Students are likely to have more confidence in the idea if the problem of ‘sufficiently good’ approximations is taken seriously.

(Can any standard school-level apparatus, in fact, detect an error of the size there appears to be between the graph and results found from s = ½at 2?)

How constant acceleration is represented in drawing the graph

See figure 3 above. In drawing the graph, each new section was drawn at a steeper slope; that is, a larger velocity. Because the acceleration was constant, the slope increased by equal amounts in each step.

See figure 4. At each moment the average velocity near that moment was worked out, using a = 10 m s-2. A line like AB in the above figure at the correct slope was put in, going an extra distance vΔt. At the next moment, v was larger, say v + Δv, where v is the extra velocity gained in an interval Δt. A line such as BC was drawn, at the new, larger slope, going a larger extra distance (v + Δv) Δt.

If, as in figure 5, AB is run on at the same slope, to D, then DF = vΔt is the extra distance that would have been covered if the velocity had not increased. The extra extra bit CD = ΔvΔt is the 'extra extra distance’ gone because the velocity did increase, by Δv.

So CD = ΔvΔt = at)(Δt) = a(Δt) 2

A rule for drawing graphs of acceleration

Looking at figure 6, there is a simple rule for drawing acceleration graphs. Take the graph AB as found at the last interval, and run on AB straight to D, as if there were no acceleration. Then add an extra extra distance DC, where DC = a(Δt) 2. Then BC is the next bit of graph. If the acceleration should vary, the extra extra distances like DC will vary; just work them out from the basic recipe.

The rule with a Δ notation

For some students, the argument in terms of acceleration will be more than enough. But others may like to see how the calculus notation is useful. Δ is used to mean small change in . . ..

Over AB, in figure 7, the velocity v is the slope of AB, that is ΔsΔt. Then the velocity rises, and the acceleration is

Over the timeΔt, the distance s changes by more than it would have changed if there had been no acceleration, and Δs in the second interval is larger than Δs in the first by the ‘extra extra distance’ Δ(Δs). Further, the acceleration is Δ(Δs)Δt 2, equal to the rate of change of velocity around B.

So the rule remains the same: add on an 'extra extra distance' over and above that which would come from taking the graph straight on. The size of the 'extra extra distance' should be:

Δ(Δs) or Δ2s = (acceleration)Δt 2.

ΔsΔt approximates to the velocity v, or dsdt.

Δ2sΔt 2 approximates to the acceleration a, or d 2sdt 2.

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