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### Acceleration

- Compensating for friction
- Distinguishing between velocity and acceleration
- Finding average acceleration with a ticker-timer
- Velocity-time graphs with a ticker-timer
- Non-uniform acceleration with a ticker-timer
- Investigating motion on a sloping surface
- Measurement of acceleration using light gates
- Using speed-time graphs to find an equation
- The automatically straight-line graph

## Acceleration

for 14-16

Experiments in this collection help students to distinguish between average and instantaneous velocities, and to understand that 'metres per second per second' is an appropriate unit for a rate of change of velocity. Visualizing velocity changes can be a vital first step, so velocity-time graphs and multiflash photos are particularly effective.

**Class practical**

It is possible to arrange for trolleys to behave as if they are moving without friction.

Apparatus and Materials

*For each student or group of students*

- Runway, with means to produce a uniform slope
- Dynamics trolley
- Elastic cords for accelerating trolley
- Rod for attaching elastic cord to trolley
- Ticker-tape
- Ticker-timer with power supply unit

Health & Safety and Technical Notes

A long runway requires two persons to carry it and set it up on the bench.

Check that a string is placed across the bottom of the runway to stop the trolley.

Read our standard health & safety guidance

Procedure

- Set up the runway with a slope that you can adjust. Place the trolley on it.
- Give the trolley a short and gentle push. If it slows down and stops that it is because of friction. If it speeds up it is because the effect of gravity is stronger than the effect of friction.
- Adjust the slope so that when you start the trolley it keeps going at a steady speed. Then the effects of friction and gravity are in balance.
- Test this with ticker-tape attached, and using the ticker-timer. This might increase the frictional effect slightly. Start the trolley and allow it to run down the slope. If it travels for most of its journey so that the dots are equally spaced, then its speed is constant.
- Attach an elastic cord to the trolley. By pulling on the elastic cord and always stretching it by the same amount you can exert a constant force on the trolley.
- Attach a new length of ticker tape. Produce a ticker-tape record of a trolley journey with a constant force.
- Use the ticker-tape record to work out the trolley's acceleration.

Teaching Notes

Students often overdo

the slope and overcompensate for friction. That is why they should use a ticker-tape record to test whether speed is constant.

*This experiment was safety-tested in March 2005*

### Up next

### Distinguishing between velocity and acceleration

**Demonstration**

This experiment allows you to focus on what happens when a constant force is applied to a trolley: its velocity will change but its acceleration will remain constant. Datalogging enables you to avoid the distraction of calculations and instantly display the acceleration.

Apparatus and Materials

- Light gate, interface and computer
- Dynamics trolley
- Mass, 200 g, pulley and string
- Clamp
- Ruler
- Double segment black card (see diagram)

Health & Safety and Technical Notes

Take care when masses fall to the floor. Use a box or tray lined with bubble wrap (or similar) under heavy objects being lifted. This will prevent toes or fingers from being in the danger zone.

Read our standard health & safety guidance

Pass a piece of string with a mass hanging on one end over a pulley. Attach the other end to the trolley so that, when the mass is released, it causes the trolley to accelerate. Choose a length of string which allows the mass to touch the ground when the trolley is about two thirds along the bench length.

Fit the trolley with a double segment black card. Clamp the light gate at a height to allow both segments of the card to interrupt the light beam when the trolley moves through the gate. Measure the width of each segment with a ruler and enter the values into the data-logging software. Connect the light gate via an interface to a computer programme.

Configure the program to obtain measurements of acceleration derived from the double interruptions of the light beam by the card. The program will require the dimensions of the card. The internal calculation within it involves using the interruption times for the two segments to obtain two velocities. The difference between these divided by the time between them yields the acceleration.

Procedure

**Data collection**

- Pull the trolley back so that the mass is raised to just below the pulley. Position the light gate so that it will detect the motion of the trolley soon after it has started moving. Set the software to record data. Release the trolley. Observe the measurement for the acceleration of the trolley.
- Move the light gate about 20 cm further along the table. Repeat the measurement, releasing the trolley from the same point as before. Observe the new measurement and compare it with the first.
- Repeat this process several times until the light gate is as near as it can be to the pulley, while still allowing the trolley to pass through.

**Analysis**

- Analysis of the data need not be treated formally.
- Students should observe that the value of acceleration does not change substantially for successive positions of the light gate. There will however be a point after which the acceleration drops close to zero. This is the point at which the mass reaches the ground, the string becomes slack, and the trolley continues to move due to inertia.
- For many students the result comes as a surprise, since they tend to associate the faster moving trolley with a greater acceleration.

Teaching Notes

- This is a useful demonstration for provoking thinking about the distinction between velocity and acceleration. The instant presentation of the acceleration, using the software, greatly benefits the discussion by avoiding preoccupation with the calculation process.
- Through direct observation with the naked eye, the velocity is easily observed to increase, but the rate of change of velocity after the mass touches the ground is much harder to perceive, especially as the trolley by then has achieved its maximum velocity. Skilful leadership of the discussion during the demonstration can sharpen pupils' concept of acceleration as a rate of change of velocity, a concept which is independent of how fast the trolley is actually moving at the time.
- The demonstration could lead into a discussion of Newton's first two laws of motion. The trolley continues to move at its maximum speed once the string goes slack. This introduces Newton's first law. The trolley only accelerates while the external force is applied to the trolley (via the string). This illustrates Newton's second law.

*This experiment was safety-tested in May 2006*

### Up next

### Finding average acceleration with a ticker-timer

**Class practical**

This a practical approach to acceleration and its definition.

Apparatus and Materials

- Trolley
- Elastic cords for accelerating trolley
- Rod for attaching elastic cord to trolley
- Ticker-timer with power supply unit
- Ticker-tape
- Sellotape

Health & Safety and Technical Notes

If trolley runways are used, remember that one is too heavy for a single adult to manipulate and carry safely.

Read our standard health & safety guidance

Procedure

- Thread a length of ticker-tape through a ticker-timer and attach the end to a trolley.
- Pull a trolley with a fixed force along a bench. Loop one end of the elastic cord around a rod attached to the trolley. Keep the force constant by making sure that the cord is always stretched by the same amount as the trolley moves. Practice doing this.
- Choose and cut through a dot near to the start of the tape. Do this when the trolley is travelling quite slowly but the dots are far enough apart to clearly distinguish one from another.
- Count ten dot-to-dot spaces and cut the tape, through a dot, again. You have cut a 'ten-tick-tape'.
- Count 40 more dot-to-dot spaces along the tape. Then cut the next 10 dot-to-dot spaces to make another ten-tick-tape.
- Draw a horizontal line, as a time axis, on a piece of paper. Glue your tapes, vertically and 10 centimetres apart, so the bottom of each tape touches this axis. This 10 centimetres represents 1 second.
- Draw a vertical axis anywhere to the left of the first tape. This is a velocity axis.
- Mark a scale, in centimetres per second, on your vertical axis. Each vertical centimetre on your axis represents 5 centimetres per second.
- Use your velocity axis to help you to work out the first velocity and the second velocity. You can call these
*u*and*v*. (Remember that u comes before v in the alphabet, just as the first velocity comes before the second.)
Work out the average acceleration of your trolley during the time between the two tapes. Acceleration is - Draw a straight line between the centres of the tops of the tapes. Draw a horizontal line from the centre of the top of the first tape. Draw a vertical line through the top of the second tape. You have made a right-angled triangle. The length of the base of your triangle, using the units of the horizontal axis, is 1 second.
- Find the gradient of the line connecting the tops of the two tapes. Measure the height of the triangle, using the units of the velocity axis, which are centimetres per second (cm/s). Divide the height of the triangle, in centimetres per second, by the base, in seconds. This gives you the average acceleration in centimetres per second per second (cm/s
^{ 2 }).

rate of change of velocity. It is equal to the change in velocity divided by the time. Average acceleration = change in velocity/time taken. The change in velocity is the difference between the two velocities,

*v – u*. The time between these two tapes is

*t*, which in this case is 1 second.

Teaching Notes

- This introduces students to mathematical modelling of accelerated motion. Many students find the concepts challenging, and the introduction in a practical context is very helpful.
- If your ticker-timers use a carbon disc, tell students to be sure to pass the tape
__under__the disc. If you are using photosensitive tape, explain that they must wait a few minutes after a run for the dots to appear. - The whole procedure above assumes that the ticker-timers produce 50 dots per second (mains frequency of the AC voltage driving them). If your ticker-timer produces more dots per second (some produce 100 dots per second) you will have to change the text.
- You will need to explain that the ticker-timer makes dots at regular intervals. You could say:
*"Each period between dots is a*tick

of time. A low voltage alternating at the same frequency as mains (50 Hz) drives the ticker-timer. If the ticker-timer produces 50 dots per second, then a tick is equal to 1/50 or ValueUnit{0.02} seconds."- Step 8 of the procedure: The length of each tape is the number of centimetres that the trolley travelled in a fifth of a second, or 0.2 s econds. So students could multiply the length by 5 to find the velocity of the trolley in centimetres per second. The actual lengths of the tapes are fixed proportions of the velocity.
- Step 9 of the procedure: The dot at the start of second tape that students make is 50 spaces away from the very first dot they chose. The time between the first dot of their first tape and the first dot of their second tape is 1 second. The time between the middle of their two tapes is also 1 second. You may need to explain this.
- An alternative, which may help students to understand the concept of acceleration, is to use mixed units for time. Ask them to calculate
*u*and*v*in cm/s, but describe the time between these tapes in 'ten-ticks'. This will give units of acceleration as 'cm/s per ten-tick'. - The procedure says nothing about the distinction between speed as a scalar quantity and velocity as a vector quantity. But since acceleration is a vector quantity we have chosen to write in terms of velocity.
- The nature of the motion in the period between the two tapes makes a good discussion point. Students should imagine the intervening tapes. Will the tapes increase steadily in length, or not? If they do not, then the gradient of the velocity-time graph varies, and acceleration varies. The intervening tape could be cut into 10 tick lengths and stuck at the appropriate position on the graph to see if their prediction was correct.
- If the tape increases steadily in length, then the gradient is constant, and the actual acceleration always matches the sloping line which students have drawn on their graphs. Acceleration is then constant, or uniform. And in that case average acceleration, and acceleration at all times during the motion, are the same. Then:
*acceleration, a = (v-u)/t*.

*This experiment was safety-tested in January 2005*

### Up next

### Velocity-time graphs with a ticker-timer

**Class practical**

Students create velocity-time graphs from their own measurements for different kinds of motion. In a qualitative way only, the significance of gradient is emphasized.

Apparatus and Materials

- Runway, with means to produce a uniform slope
- Dynamics trolleys
- Tricker-timer with power supply unit
- Ticker-tape
- Sellotape

Health & Safety and Technical Notes

Read our standard health & safety guidance

Trolley runways are heavy. Do not expect one technician to be able to manipulate and carry one unaided. If the load presented by one end is less than 10 kg , one runway could be carried by two 15-year olds.

Procedure

**Accelerating down a slope**

- Set up the runway with a significant slope, with a buffer at the bottom to catch the trolley.
- Test what happens when you put a trolley on it. It should accelerate down the slope.
- Thread a length of ticker-tape, about the same length as the trolley's journey down the slope, through the ticker-timer. Stick the end of the tape to the trolley.
- Turn the ticker-timer on, and allow the trolley to accelerate down the slope.
- Cut the tape through the dot (or set of overprinted dots) produced just before the trolley was released. Count ten dot-to-dot spaces and cut the tape again. (If the dots are too close together to distinguish them, then you will have to estimate the ten spaces.) Starting from your last cut, count ten more spaces, and cut again. Repeat this, until you have a collection of consecutive tapes, each one longer than the one before it. Number the order of your tapes, from 1 onwards.
- Draw a horizontal line on a sheet of paper. Make a 'bar chart' by sticking the tapes vertically side by side, so that their bottoms just touch the horizontal line. The first and shortest tape should be at the left hand end of the line.
- The horizontal line acts as a time axis, with its left end time zero. Mark your horizontal axis in intervals of 0.2 seconds, and add a suitable axis label.
- Draw a vertical line through the zero of the time axis. This vertical line is the velocity axis. When time was zero for the trolley journey, velocity was zero. (The trolley started from standstill, or rest.)
- Mark the scale on the vertical axis, in centimetres per second. Each vertical centimetre represents 5 centimetres per second of speed.
- Draw a smooth line through the top-centre of each tape on your velocity-time graph. Describe in words what the graph says about the trolley motion.
- Reduce the slope of the runway. Adjust it so that, when you give the trolley a brief gentle push, it travels for a large part of the runway length without slowing down or speeding up. The forces of gravity and friction are now balanced.
- Produce a set of ten-tick-tapes for a steady velocity (or approximately steady velocity) journey of the trolley.
- Make a velocity-time chart as before, using at least five consecutive ten-tick-tapes for a section of the journey for which velocity is constant. (Ignore the first tape, or tapes, when the trolley was accelerating as you pushed it. Ignore tapes for later motion if there is a lot of change in velocity.)
- Move the ticker-timer, and the trolley, to the lower end of the runway. Give the trolley a short push, so that it moves up the slope. Predict what the velocity-time graph for this motion will look like.
- Test your prediction by making a velocity-time graph with a suitable set of ten-tick-tapes. (As for the previous case, ignore lengths of tape that were made while you were pushing. Choose time zero as some time soon after your push.)
- Draw a smooth line through the top-centre of each tape on your velocity-time graph. Describe in words what the graph says about the trolley motion.

**Constant velocity on a slope**

**Motion up a slope**

Teaching Notes

- There is a lot of activity here, and you may need more than one lesson.
- If your ticker-timers use a carbon disc, tell students to be sure to pass the tape under the disc. If you are using photosensitive tape, explain that they must wait a few minutes after a run for the dots to appear.
- The whole procedure above assumes that the ticker-timers produce 50 dots per second (mains frequency of the AC voltage driving them). If your ticker-timer produces more dots per second (some produce 100 dots per second) you will have to change the text.
- You will need to explain that the ticker-timer makes dots at regular intervals. You could say:
- Step 9 of the procedure: The length of each tape is the number of centimetres that the trolley travelled in a fifth of a second, or 0.2 s econds. So students could multiply the length by 5 to find the velocity of the trolley in centimetres per second. The actual lengths of the tapes are fixed proportions of the velocity.
- Once students have completed their first velocity-time graph, discuss what it means. The bigger the acceleration, the steeper the slope of the graph. The gradient of the line is always equal to the value of the acceleration. If the line is straight, then the acceleration is constant (or
uniform

). If it curves then acceleration varies. - More advanced students could find the gradient of the graph quantitatively. If the graph is curved, they should draw tangents and find the gradient at more than one point.
- When finding gradients, big triangles produce more accurate results (and they are easier to measure).
- You might ask students how to use the tapes to construct distance-time (or displacement-time) graphs. The cumulative lengths of the tapes show total distance travelled.
- An important point is that the same motion can be represented in two related but significantly different ways, by distance-time (or displacement-time) as well as velocity-time graphs.
- The second velocity-time chart shows a trolley moving at constant velocity. The top of the tape chart is horizontal and the gradient is zero, indicating there is no acceleration.
- The third velocity-time chart shows a trolley decelerating up the slope. The gradient of the chart is negative.

*"Each period between dots is a*

tickof time. A low voltage alternating at the same frequency as mains (50 Hz) drives the ticker-timer. If the ticker-timer produces 50 dots per second, then a tick is equal to 1/50 or 0.02 s econds."

*This experiment was safety-tested in December 2004*

### Up next

### Non-uniform acceleration with a ticker-timer

**Class practical**

Students can explore another kind of motion and its velocity-time graphs.

Apparatus and Materials

*For each student group*

- Lengths of loose chain (at least 1m long)
- Ticker-timer with power supply unit
- Ticker-tape
- Sellotape

Health & Safety and Technical Notes

An empty cardboard box on the floor to catch the chain should ensure that no one's foot is at risk.

Read our standard health & safety guidance

DIY and hardware stores sell lengths of chain.

Procedure

- Put the length of chain on a smooth table so that it is at right angles to the edge. Then pull the end over the edge until, on release, the whole chain slides. Then the hanging portion will pull the rest, with increasing acceleration.
- Repeat this with a length of ticker-tape attached to the chain.
- Use the ticker-tape to create a velocity-time chart or graph.
- Draw a smooth curve to fit the top-middle of each length of tape on the chart. Draw tangents to the curve at two points. Work out the gradients of the curve at each point, and so also the accelerations.
- For an explanation of drawing ticker-timer acceleration charts see:
Velocity-time graphs with a ticker-timer

Teaching Notes

- The acceleration of the chain depends on the force pulling the chain over the edge of the table. The force increases as the length of the chain over the edge increases.
- The experiment shows that acceleration is not always constant. In fact acceleration can change and itself have a rate of change. The rate of change of distance (or, more strictly,
displacement

) is called velocity. The rate of change of velocity is called acceleration. The rate of change of acceleration has no name so it's hard to get hold of the concept. - It is interesting to investigate the motion of sets of chains with different mass/length ratios.

*This experiment was safety-tested in January 2005*

### Up next

### Investigating motion on a sloping surface

**Demonstration**

When free to move, a trolley on an inclined plane will accelerate down the slope. If it is given a velocity up the slope it will decelerate. This experiment allows both acceleration and deceleration to be investigated by datalogging the output from a motion sensor.

Apparatus and Materials

- Motion sensor, interface and computer
- Dynamics trolley
- Runway
- Wooden block and clamp

Health & Safety and Technical Notes

If a full length runway (2 m long) is used, two people should handle it to avoid back strain.

Read our standard health & safety guidance

Prop up the runway at one end to create an inclined plane. Clamp a wooden block at the lower end to make a solid barrier.

The dynamics trolley with its spring plunger pointing forwards is allowed to roll down the runway.

Place a motion sensor at the top of the runway and connect it via an interface to a computer.

Take care that the girders to the sides of the runway do not protrude too high. This would interfere with the ability of the sensor to distinguish sonic reflections from the trolley.

Configure the data-logging software to measure the distance of the trolley from the sensor and present the results as a graph of distance against time.

Procedure

**Data collection:**

- Position the trolley near the top of the runway, about 30 cm from the motion sensor.
- Start the data-logging software and release the trolley. The trolley should bounce off the wooden block and move up and down the runway several times. The distance travelled by the trolley reduces after successive bounces.
- Depending upon the software, the results may be displayed as a distance versus time graph whilst the experiment proceeds. Observe the shape of the graph as a succession of half loops.
- Identify on the graph:
- The points corresponding to the collision of the trolley with the block.
- The parts of the graph representing the trolley movement down the runway.
- The parts of the graph representing the trolley movement up the runway.

- There are several interesting features of the graph which invite explanation:
- The graph shows curves rather than straight lines.
- Successive half-loops reduce in size.
- Each half-loop is asymmetrical.
- Distances are from the sensor rather than from the wooden block where collisions occur.

- You could study the velocity of the trolley more directly. Do this by setting the program to calculate and plot the rate of change of distance against time. This reveals both positive and negative velocities.
- These change linearly, indicating uniform acceleration and deceleration. Note that the magnitude of the deceleration is slightly larger than the acceleration.

**Analysis:**

Teaching Notes

- This experiment illustrates the value of rapid collection and display of data in assisting thinking about the phenomenon under investigation. Data is collected within a few seconds and the graph is presented simultaneously. This helps students to make connections between features on the graph and the actual motion of the trolley. In general, it is a useful teaching strategy to ask students to make a prediction about the appearance or shape of a graph, before the program actually plots the result. Comparing the result with the prediction is a simple ploy for prompting discussion.
- The collisions show up as rapid reversals in the direction of motion. One side of the loop corresponds to the downward motion and the other side corresponds to the upward motion. These associations may be confirmed by separate informal experiments pushing the trolley up and down the runway by hand. Similar hand-controlled experiments can confirm the connection between the speed of the trolley and the gradient of the graph.
- The shape of the graph is such that it shows distance increasing as the trolley rolls
down

the runway. For some students this is counter intuitive. They may think thatupward

graph line indicatesupward

movement on the runway. Emphasize that the graph shows notvelocity

butdistance

, which is measureddown

the slope. - Software tools for taking readings from the graph and measuring gradients at several points are useful for testing out ideas about the motion. For example: 'Is the gradient immediately before the collision the same as the gradient immediately after?' This prompts thinking about the change of velocity at each collision. The asymmetry of each loop can also be investigated. Further experiments can establish that the degree of asymmetry depends upon the mass of the trolley and the angle of slope. The presence of friction has a role in explaining the asymmetry, as discussed below.
- A particularly useful software function is to calculate the velocity for all points on the graph and plot these as a new graph. The linear changes observed confirm uniform acceleration and deceleration. There are plenty of opportunities for discussion here.
- The gradients of the velocity versus time graph are observed to be different according to the direction of travel.
- The
deceleration

when the trolley is moving up the slope is found to be slightly larger than theacceleration

when the trolley is moving down the slope. For an explanation of this, prompt students to think about the direction of the force of friction and the component of its weight acting along the ramp. - For the downward motion, friction opposes the component of gravity. For the upward motion friction is in the same downward direction as the component of gravity. The resulting force and acceleration are different in each case.

**How Science Works Extension:**

- This experiment provides ample opportunities for extension work, illustrating various aspects of ‘How Science Works’. For example, students could investigate how the acceleration of the trolley depends on its mass. In principle, the mass of the trolley is irrelevant; in a frictionless situation, the trolley will have an acceleration equal to the component of g down the slope. However, friction reduces the acceleration and the effect is greater for a trolley with a small mass.
- Because the electronic equipment can determine the value of the acceleration, students are able to focus their attention on the experiment rather than on the manipulation of ticker-tapes or on complicated calculations.
- Frictional effects are more significant for a ramp with a small angle of slope, so students should consider this in the design of their experiment.
- Because the effects of friction are small, students will need to work carefully, making repeat measurements if they are to show up any clear effects. This is a good test of experimental technique.

*This experiment was safety-checked in May 2006*

### Up next

### Measurement of acceleration using light gates

**Demonstration**

This activity uses light beams and timing technology to obtain values for acceleration.

Apparatus and Materials

- Scaler or datalogging timing system
- Light beam assemblies (sources and sensors), 2
- Dynamics trolley
- Runway, with means to produce a uniform slope
- Elastic cords with means of fixing them to the trolley, 3
- Card
- Stopwatch or stopclock

Health & Safety and Technical Notes

A long runway is too heavy for one person to carry and manipulate: ensure that two persons are available to set it up.

Read our standard health & safety guidance

Whichever device you use, you will need to know how it operates.

The card should be 10 or 20 cm long and 5 or 6 cm wide.

Depending on the timing device, the lamps and sensors may be connected separately to the timer, or both of them connected in series with the timer.

Procedure

- Put the trolley on the runway and adjust the slope so that, when you give the trolley a push, it rolls at a near-steady speed. The force of gravity now compensates for the effects of friction. The trolley behaves approximately as if there were no friction.
- Set up the two light beam source and sensor pairs, at least one metre apart, along the runway near to the beginning and end of the trolley's journey.
- Fix the card to the trolley so that it breaks the light beam between both source and sensor pairs as it moves down the runway.
- Pull the trolley along the runway, starting from above the first light beam, with a constant force. Use a single elastic cord to do this, keeping it at a fixed amount of stretch or extension. (Aligning the end of the cord with the front edge of the trolley enables you to keep it at constant length.)
- Measure the times for which the card is blocking each of the light beams. If you are using a scaler then you will need to read the first time before the trolley gets to the second beam. If you are using a datalogging system then it will record both of the times.
- The distance travelled by the trolley during the two measured times is the width of the card. Divide distance by time in each case to work out the two velocities.
- To work out acceleration, you need to know the time that the trolley takes to travel from one beam to the other. If you are using a scaler system, you will need to use a separate stopwatch or stopclock for this. Your datalogging system may be capable of measuring the time directly.
- Use
*a = (v - u)/t*to find the trolley's acceleration, where*a*= acceleration,*v*= final velocity,*u*= initial velocity,*t*= time between*u*and*v*.

Teaching Notes

- This measurement of acceleration takes little time once you have set up the apparatus. Let the students take their own measurements in groups of two or more.
- You can extend the activity beyond the simple measurement of acceleration to an investigation of the dependence of acceleration on applied force or on mass. Force can be varied simply by adding more elastic cords in parallel. Each extra cord, all of them always stretched by the same amount, adds an extra
unit

of force. You can vary mass by stacking trolleys one above the other, using the metal rods supplied. - The activity raises issues of the distinction between average and instantaneous values of velocity and acceleration. If the cards were of vanishing width, they would give values of distance and time for a single instant. This is of course impossible, but the resulting calculated velocity would be the velocity at that single instant. (Note that a speedometer
does

supply values of practically instantaneous velocities, so it is not merely a theoretical concept.) Since the card has a finite width, the recorded values of distance and time give the velocities averaged over the fairly short times for which the card breaks the beams. - Even with instantaneous values of velocities, this experiment yields information about average acceleration during the time between the two measurements. It does not measure the acceleration instant by instant as the trolley travels. However, since force and mass are constant, acceleration is constant. Thus the average acceleration and its acceleration at any instant are, in fact, the same.
- If the timing device does not appear to count correctly:
- Try adjusting the relative position of the lamp and detector to get maximum illumination of the latter
- Try reversing the polarities of the detector.

*This experiment was safety-tested in December 2004*

### Up next

### Using speed-time graphs to find an equation

Imagine a graph plotted with SPEED on the vertical axis against TIME on the horizontal axis.

#### Constant speed

For an object moving along with constant speed *v*, the graph is just a horizontal line at height *v* above the axis. You already know that *s*, the distance travelled, is speed multiplied by time, *vt*; but on your graph *v* x *t* is the AREA of the shaded block of height *v* and length *t*.

#### Constant acceleration

Sketch a graph for an object starting from rest and moving faster and faster with constant acceleration. The line must slant upwards as *v* increases. And if the acceleration is constant the line must be a straight

slanting line.

Take a tiny period of time from *T* to *T'* on the time axis when the speed was, say, *v*_{1} . Look at the pillar that sits on that and runs up to the slanting graph line (Graph III). The area of that pillar is its height *v*_{1} multiplied by the short time *TT'*. That area is the distance travelled in that short time.

How big is the distance travelled in the whole

time, *t*, from rest to final *v*? It is the area of all the pillars from start to finish. That is the area of the triangle (in Graph IV) of height final *v* and base *t*, the total time.

The area of any triangle is ½ (height) x (base).

So distance *s* is ½(height, *v*) x (base, *t*) *s* = ½*v**t*.

Suppose the object *does not start from rest* when the clock starts at 0 but is already moving with speed *u*. It accelerates to speed *v* in time *t*. Then the graph is like graph V below; and the distance travelled is given by the shaded area. That is made up of two patches, a rectangle and a triangle (Graph VI).

The rectangle's area is *u**t*, the triangle's is ½(*v*-*u*)*t*.

Then *s* = *u**t* + ½(*v*-*u*)*t*

- =
*u**t*+ ½*v**t*-½*u**t* - = ½
*v**t*+ ½*u**t* - = (
*v*+*u*2)*t*

Alternatively, since *v*-*u* = *a**t*

*s*=*u**t*+ ½(*v*-*u*)*t*can be expressed as- =
*u**t*+ ½(*a**t*)*t* - =
*u**t*+ ½*a**t*^{ 2}

These formulae are only true for constant acceleration. Look at Graph VII. Is the acceleration constant? Which part of the area for *s* is different now?

What part of *u**t* = ½*a**t*^{ 2} is no longer safe for calculating *s*?

### Up next

### The automatically straight-line graph

Examples of using a straight line graph to find a formula.

** Example 1: To show that π R^{ 2} gives the area of a circle. **

For any circle π is the number 3.14 in the equation:

circumference = 2π x radius or π x diameter

So π is circumferencediameter

Starting from that (as a definition of π) we can show that the area of a circle is π*R*^{ 2}.

Draw a large circle with centre 0 and radius *R*. Plot a graph of 2π*r* upwards against *r* along.

Then the graph must

be a straight line and its slope

will be 2π.

The end-point A, of the graph belongs to a big circle of radius *R*. Each other point of the graph: line 0A belongs to a smaller circle, of radius *r*.

Sketch III shows two small circles close together with radii (*r*) and (*r* + tiny bit of radius).

What is the area

of the shaded ring between them? The ring has width (tiny bit of radius) and length

2π*r* (its circumference). Its area

is 2π*r* x (tiny bit of radius).

On the Graph IV the shaded pillar shows just that same area

, 2π*r* x (tiny bit of radius).

Now ask about all such rings from the centre 0 out to radius *R*. Their total area is the same as the area of all the pillars in Graph V. That is the triangle of height2π*R* and base *R*.

AREA = ½2π*R* x *R* = π*R*^{ 2}.

Therefore area of circle is π*R*^{ 2}.

** Example 2: To show that s = ½at^{ 2} for constant acceleration from rest. **

Plot a graph of at upwards against *t* along. Then with a constant the graph must

be a straight line; and its slope will be *a* (Graph VI).

Choose a tiny bit of time on the t-axis and draw a pillar up to the line (Graph VII). The area of the pillar is: height x width,

(*a**t*) x (tiny bit of time)

and that is (*v*) x (tiny bit of time), since acceleration x time **is** speed.

And that is (tiny bit of time travelled).

Then total distance travelled, *s*, is given by the total area of all such pillars (Graph VIII).

*s* = area of triangle OAB = ½*a**t* x *t* = ½ *a**t*^{ 2}.