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## Kinematics

Lesson for 16-19

In this topic, students learn to describe motion in terms of displacement, velocity, acceleration and time. There is only a limited discussion of the forces involved; kinematics is concerned with describing motion, while dynamics is concerned with explaining motion in terms of the forces acting.

## Episode 204: Preparation for kinematics

Teaching Guidance for 16-19

- Level Advanced

There are opportunities here for using datalogging equipment and light gates to record and analyse motion, so it would be useful to check in advance what is available and try it out. One approach with unfamiliar equipment is to ask a couple of students to set up an experiment or demonstration in advance, so that they can then demonstrate it to the class (under your guidance, of course). In this way, you can learn alongside them.

Similarly there are opportunities for using modelling software (such as Modellus). Try finding some students who are prepared to try it out and give you a driving lesson.

## Main aims of this topic

Students will:

- Understand and use the relationships between velocity, acceleration, time and displacement
- Know and use the following equations of motion:
*s*=*u**t*+ 12*a**t*^{ 2},*v*=*u*+*a**t*,*v*^{ 2}=*u*^{ 2}+ 2*a**s* - Calculate acceleration from the instantaneous gradient of a velocity time graph
- Calculate displacement from the area under a velocity-time graph
- Understand the independence of horizontal and vertical velocities
- Describe motion in a uniform gravitational field using the independence of vertical and horizontal velocity
- Use the equations of motion to calculate ranges of horizontally projected bodies

## Prior knowledge

Students will be familiar with calculations of speed and (perhaps) acceleration. It will help if they are familiar with laboratory methods of determining times, speeds and accelerations using light gates and datalogging equipment.

## Where this leads

Explaining motion in terms of forces is covered in the topic of dynamics.

### Up next

### Describing motion

## Episode 205: Describing motion

Practical Activity for 16-19

- Activity time 75 minutes
- Level Advanced

This episode introduces (or revises) basic kinematics. The equations of motion are met in separate science (triple award) pre-16 level but are not discussed in detail in double award. This is an area which many moderate-to-weak students find difficult and they may well remember their pre-16 experience with some unease. It is worthwhile including a direct measurement activity to ensure that students spend sufficient time on the basic physics of velocity, distance and time. This is a busy episode that focuses on one activity that all students should perform individually.

Lesson Summary

- Discussion: Scalars and vectors, velocity and displacement (5 minutes)
- Student experiment: Balls down ramps (25 minutes)
- Discussion: Average velocity, time and displacement (10 minutes)
- Discussion: Chalk in the air (5 minutes)
- Student questions: Using these ideas (30 minutes)

## Discussion: Scalars and vectors, velocity and displacement

If necessary, briefly remind students of the difference between vectors and scalars. Taking a marble and rolling it along the bench in one direction then back along the bench in the other direction at the same speed clearly demonstrates the difference between velocity and speed. Defining acceleration as the rate of change of velocity allows a discussion of an orbiting body (or a conker on a string) moving with constant speed whilst also experiencing constant acceleration.

Before proceeding with the student experiment it is worthwhile explaining why the final velocity of a body is twice its average velocity if (and only if) it has uniformly accelerated from rest.

## Student experiment: Balls down ramps

Rolling balls down ramps: This is a version of the experiment that Galileo performed towards the end of the sixteenth century. Students time a marble rolling a fixed distance down the slope. From this, they can produce graphs of final velocity against distance and final velocity against time. The first graph will show a square relationship whereas the velocity-time graph will show a linear relationship. However, there is much scatter in the points and it is a useful exercise to consider whether the velocities are more accurate

for readings over small distances than those over larger distances.

Episode 205-1: Rolling balls down ramps (Word, 39 KB)

## Discussion:Average velocity, time and displacement

Using the graphs drawn from the experiment you can show two things:
*a* = Δ *v* Δ *t*
that is, that the gradient of a velocity-time graph yields acceleration.

The area under the velocity time graph gives the distance travelled by the marble.

Ask your students to read off a time (say 1.5 s) from the distance-time graph and find the corresponding distance value. The area under the velocity-time graph up to 1.5 s will come to the same value as the distance. Bright students will see the circularity in this method, but this, in itself, is of value.

## Discussion: Chalk in the air

Throw a piece of chalk (or a board marker) in the air vertically and catch it when it returns to its original position. Ask students to sketch a velocity-time graph of the motion, assuming no air resistance.

Very few will produce a graph like this:

The errors made by the students will allow you to highlight many points of physics. For instance, a graph may suggest that the acceleration due to gravity is different when a body is travelling upwards from the value for a descending body. Particularly revealing is to ask what is the acceleration when the object reaches its maximum height. You will find this a fruitful discussion from which even the bright and confident students can learn a lot.

## Student questions: Using these ideas

Select from the questions those most suitable to the level of your students. You might choose to give different questions to different students – for example, those not taking post-16 level mathematics might need to work through more examples than those studying mathematics.

Episode 205-2: Distance, time and speed calculations (Word, 63 KB)

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### Up next

### Uniform and non-uniform acceleration

## Episode 206: Uniform and non-uniform acceleration

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

This episode continues to look at basic kinematics and introduces the equations of motion for uniform acceleration. This involves a little calculation practice.

Lesson Summary

- Demonstration (or student experiment): Non-uniform acceleration (20 minutes)
- Discussion: Developing equations of motion (10 minutes)
- Student experiment: Measuring acceleration due to gravity
*g*(20 minutes) - Student questions: Calculations (30 minutes)
- Worked example: Average velocity (10 minutes)

Uniform acceleration is compared with non-uniform acceleration.

## Demonstration (or student experiment): Non-uniform acceleration

Students will have already considered uniformly accelerated motion. This demonstration (or experiment) uses a similar method to consider non-uniform motion. You can conclude the demonstration by discussing the relationships shown on the graphs, stressing that these hold for both uniform and non-uniform motion.

Episode 206-1: An experimental velocity-time graph (Word, 35 KB)

## Discussion: Developing equations of motion

Here you can develop the equations of motion (the SUVAT equations

).

Confident mathematicians will enjoy the mild challenge of developing SUVAT equations whereas weaker or non-mathematical students may find the activity surprisingly difficult. It is therefore best to proceed through the activity at a reasonable pace so that you concentrate on the results and using the SUVAT equations.

Episode 206-2: Deriving the equations of motion (Word, 41 KB)

## Student experiment: Measuring acceleration due to gravity g

Measuring the acceleration due to gravity g is a nice, simple experiment that also brings up the concepts of precision and accuracy. Of course, the students will know

the value of *g*, and may well have measured it. Nonetheless, it is a useful exercise to build good experimental practice.

Episode 206-3: Measuring the acceleration of free fall (Word, 21 KB)

## Student questions: Calculations

These are a few simple questions that go over the ideas met in the unit. They include practice with interpreting motion graphs.

Episode 206-4: Motion under gravity (Word, 38 KB)

Episode 206-5: Thrust SSC (Word, 67 KB)

## Worked examples: Average velocity

You might like to use the question below to highlight that the equations of motion (SUVAT equations) only apply to uniform acceleration.

A cyclist travels a displacement of 300 m due North at a velocity of 10 m s^{-1}. She travels the next 300 m in the same direction at a velocity of 15 m s^{-1}. Calculate the average velocity of the cyclist.

Answer: First 300 m takes:

300 m10 m s^{-1} = 30 s

Second 300 m takes:

300 m15 m s^{-1} = 20 s

average velocity = total displacementtotal time

average velocity = 600 m50 s

*v*_{average} = 12 m s^{-1}

Many weaker pupils will assume the answer is 12.5 m s^{-1} . You will have to explain why the equation:

average velocity = *v*+*u*2

cannot be used in this example. The equation only applies to uniformly accelerated motion. The cyclist spends longer travelling at 12 m s^{-1} than at 15 m s^{-1} .

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### Up next

### Projectile motion

## Episode 207: Projectile motion

Lesson for 16-19

- Activity time 85 minutes
- Level Advanced

This episode looks at the independence of vertical and horizontal motion. It concerns objects accelerating vertically when projected horizontally or vertically. The crucial concept is that vertical acceleration does not affect horizontal velocity. This explains all projectile motion. You can discuss why this is the case dynamically – but this is best left until later in the study of mechanics. Getting the basic concept across should be your priority. Similarly, make it clear that you are ignoring any effects of drag at this stage.

Lesson Summary

- Demonstration and discussion: Motion in a parabola (10 minutes)
- Demonstration: Monkey and hunter (10 minutes)
- Demonstration: Pearls in air (5 minutes)
- Student investigation: Range of a projectile (30 minutes)
- Student experiment: Gravity and archery (30 minutes)

(Note that a projectile is an object which is initially projected by a force, but which then continues to move freely under the influence of gravity; a rocket which is firing its motors is not a projectile.)

## Discussion and demonstrations: Motion in a parabola

Here are two quick demonstrations showing motion in a parabola.

The first is a quick, fun demonstration that focuses the students’ minds on parabolic motion. This is something we all know

but this episode is all about explaining the motion. Follow this with the diluted gravity

demonstration.

Episode 207-1: A thrown ball follows a parabolic path (Word, 25 KB)

Episode 207-2: Diluted gravity – projectile paths (Word, 59 KB)

Having successfully obtained a parabola the following tasks can be used to move the students’ understanding forward:

Describe the motion, as precisely as possible, in words. (No hand waving!)

If this proves difficult try breaking up the motion into horizontal and vertical components. What is happening to the vertical velocity? (It’s decreasing, then changing direction and increasing, i.e. vertical acceleration.) What about the horizontal velocity? (It’s constant.)

Further pointers: Ignoring air resistance, is anything resisting the horizontal motion? (No!)

Will the acceleration due to gravity be different for a horizontally moving object? (No, again!)

You can use Multimedia Motion to generate a graph of projectile motion which clearly shows the independence of horizontal and vertical velocities.

The discussion should develop the ideas of independence of horizontal and vertical motion and uniform horizontal velocity and uniform vertical *acceleration*. Hence, horizontal and vertical displacements are given by:

*s*_{horizontal} = *v*_{horizontal} × *t* and

*s*_{vertical} = *u*_{vertical}*t* + 12*a**t*^{ 2},

In many cases *u*_{vertical} is zero.

## Demonstration: Monkey and hunter

The idea that vertical and horizontal motions can be considered separately is demonstrated in a dramatic fashion in the classic monkey and hunter

experiment.

You need to set this up in advance and check that it is working. When it does work it is a superb illustration. Questioning should focus on why this demonstrates independence of horizontal and vertical motion – both objects have fallen the same distance in the same time.

Episode 207-3: Mid-air collisions (Word, 56 KB)

## Demonstration: Pearls in air

Water droplets also follow a parabolic path in the air. This gives another clear demonstration of the effect. Again, concentrate on the explanation in terms of independence of horizontal and vertical motions.

Episode 207-4: Pearls in air (Word, 45 KB)

## Student investigation

Here is an interesting approach to projectile motion in which students fire a marble towards a target. This gives students, working independently or in pairs, an opportunity to design a simple experiment that will give them practice using the SUVAT equations.

Episode 207-5: Build and test a marble launcher (Word, 77 KB)

The students should begin by considering vertical motion. If they set their launcher *x* metres above the sand pit, the time the ball will be in the air can be found from:
*s*_{vertical} = *u*_{vertical}*t* + 12*a**t*^{ 2}
where
*s*_{vertical} = *x*
and
*u*_{v} is 0;
hence
*t* = *2 x*

*g*From the horizontal range, the horizontal velocity can be calculated. This value can be used to predict the range when the height above the pit is changed to 2*x* , 3*x* , 4*x* and so on.

A table can be constructed with the following headings:

Height / m. Calculated time in air / s. Predicted range / m. Measured range / m.

This can be completed for homework – it gives useful practice in SUVAT. Graphs can be constructed of predicted and measured ranges against height. Students should comment on the comparison between predicted and measured ranges.

## Student experiment: Gravity and archery

Keen students can extend the investigation to consider the effect of gravity in the sport of archery.

Episode 207-6: Challenges in target archery (Word, 22 KB)