Additional Physics Topic 2 - Forces and Motion

Force and Acceleration

For a car, the forward force is the force from the engine. The backwards forces are:

• air resistance;

• friction from the road, and the drive-train (gearbox, wheels, etc.).

When forces are balanced, the resultant force is zero. A vehicle that has balanced forces acting on it will stay stationary, or at a constant speed.

If there is a resultant force, the vehicle will accelerate in the direction of the resultant force. The acceleration will be dependent on the direction of the force:

• If the force is in the same direction as the movement, the car will go faster.

• If the force is in the opposite direction, the car will slow down.

• If the force is at 90o to the direction of the movement, the car will go round a corner.

The bigger the force, the bigger the acceleration.  If we use the same force for different masses, the bigger the mass, the less the acceleration.  The van and the car have exactly the same engine and gives out exactly the same force. The car accelerates a lot faster.  (These cars are looking dated now.)

A common measure of acceleration is the "0 - 60 mph" figure (60 mph = 27 m/s). That has now been changed to the "0 - 62 mph" figure, i.e. the 0 - 100 km/h figure, which is a time in seconds. For these two vehicles it is:

• For the Mondeo, 10 s;

• For the Transit van, 15 s.

We can work out the acceleration by using a simple equation:

acceleration (m s-2) = 27.8 ÷ (0 - 100 km/h time)

 Question 1 100 km/h = 27.8 m/s.  The Mondeo has a 0 - 100 km/h time of 10.5 seconds.  What is its acceleration in m s-2? We can investigate force and acceleration by pulling a physics trolley with a force meter, and measuring the acceleration with ticker tape. It is not an easy experiment from which to get reliable data, and the working out of the data is tedious. You can, of course, use a motion sensor with a computer.

What we do find is that:

For a constant mass, the acceleration is directly proportional to the force.

This is shown in the graph: The graph starts at the origin (zero force, zero acceleration). If the force doubles, the acceleration doubles too.   We say that force is directly proportional to acceleration.

 Question 2 Use the graph to explain what is meant by the phrase "directly proportional". This important finding is called Newton's Second Law of Motion.

a µ F

If we change the mass, and plot the data on a graph, we see that the graph is a curved graph called a hyperbola: The graph shows that the acceleration is inversely proportional to the mass. Double the mass, and the acceleration is halved.

So we can state that:

a µ 1/m

We can sum this up with the important equation:

Force (N) = mass (kg) × acceleration (m s-2)

In physics code:

F = ma

In triangle form: The unit for force is Newton (N), which is defined as:

the force needed to accelerate a 1 kg mass at a rate of 1 m s-2

The unit m s-2 is said as "metres per second squared", or "metres per second every second".

 Worked Example A train of total mass 45000 kg has a drag force of 10000 N  the locomotive can produce a forward force of 40 000 N. (a) What is the resultant force? (b) What is the acceleration of the train? Answer (a) Resultant force = forward force - drag = 40 000 N - 10 000 N = 30 000N   (b) Rearrange: a = F/m = 40 000 N ÷ 45 000 kg = 0.89 m/s2

 Question 3 (a) The Mondeo has a mass of 1400 kg.  Use your answer to Question 1 to work out the force from its engine. (b) Assume that the engine in a Transit van gives out the same force as the engine in the car.  If the Transit has a mass of 2300 kg, what is its acceleration? Distance, Speed, and Time

Linear motion describes how objects travel in a straight line.  There are three quantities that we can measure:

• Speed, physics code s, in metres per second - the rate at which it covers distance;

• Distance, physics code d, in metres;

• Time, physics code t.

You will have used the familiar equation:

Speed (m/s) = distance (m)

time (s)

In triangle form: In these notes we will use SI units (m/s) although you can use the formula with km per hour, or miles per hour.  Some useful conversions are:

• 1 km/h = 0.28 m/s

• 1 mph = 0.44 m/s

 Question 4 A runner completes a 100 m race in 9.8 seconds.  What is her average speed? Velocity is speed in a given direction, e.g. 4 m/s from left to right.  This example will show the difference.  Suppose the runner travels 63 m by running round the track.  If you look at the straight line distance between A and B, you will see that it is 40 m.  The runner's velocity will be given by

velocity = straight line distance from A to B ÷ time Question 5 A runner runs half way round the race track above.  He runs a total distance of 63 metres at a speed of 8 m/s. (a) What time does he take? (b) What is his velocity from A to B?  (It is NOT 8 m/s)  Velocity is NOT a posh word for speed.

Acceleration

Acceleration is the rate at which speed changes:

• When objects go faster, the acceleration is positive;

• When objects slow down, their acceleration is negative (deceleration).

The units for acceleration look strange - metres per second squared (m/s2).  It is easier to say this as "metres per second every second".

The equation that links speed and acceleration is:

acceleration (m/s2) = change in speed (m/s)

time (s)

In triangle form: The physics codes are:

• a - acceleration

• t - time

• Ds - change in speed.  The triangular symbol D is "Delta", a Greek capital letter 'D', which means "change in".  You will see this used a few times at GCSE, but more at A level.

The change in speed can be worked out easily:

Change in speed = speed at end - speed at start

 Worked Example 60 mph is 27 m/s.  What is the acceleration of a car whose 0 - 60 figure is 13.5 s? Answer First of all work out the change in speed. Ds = 27 m/s - 0 m/s = 27 m/s Acceleration is now worked out: a = 27 m/s ÷ 13.5 s = 2.0 m/s2

 Question 6 A car accelerates from 20 m/s to 40 m/s in 15 s.  What is its acceleration? If the speed at the end is lower than the speed at the start, the acceleration is negative.

 Question 7 A car slows down from 30 m/s to 15 m/s in 10 s.  What is its acceleration? Motion Graphs

We can represent the movement of objects using a graph, usually plotting time on the x-axis (horizontal) and the speed or distance on the y-axis (vertical).

Distance-Time Graph

Let us have a look at a distance time graph of a cyclist: Between A and B the cyclist is cycling at a constant speed.  The gradient (slope) of the graph gives us the speed.  Since the gradient is a straight line, the speed is constant.

The area under the distance-time graph has no meaning.

 Question 8 What is happening between B and C, and C and D? This graph shows another distance time graph, this time with acceleration. Question 9 How would you tell from a distance-time graph that an object was accelerating? Finding Speed from a Distance-Time Graph  (Higher Tier Only)

The gradient of the distance-time graph is the speed.  If the graph is of displacement against time, the gradient is velocity.

To work out the gradient, you need to measure the rise and the run. Speed = gradient = rise ÷ run

 Question 10 Use the gradient of this sketch graph to calculate the speed of this object.  If the graph is curved, you have to draw a tangent to the curve, then work out the gradient of the tangent.  (This graph is a bit wobbly, but the idea is the same.) Speed-Time Graph

The graph below shows a train accelerating from a station along a straight and level track to a maximum speed and slowing down to a stop at the next station.  The easiest way to show this is with a speed time graph. Acceleration is the gradient of the speed-time graph.

From the graph,

• between O and A, the train is accelerating;

• between A and B, the train travels at a constant speed;

• between B and C, the train slows down.  Slowing down can also be called negative acceleration, or deceleration.  It is given a minus sign.

Distance is the area under the speed-time graph.  To work out the total distance, we would add the areas of:

• triangle OAX;

• rectangle ABXY;

• triangle BCY.

At GCSE acceleration is uniform, which means that the speed [velocity] is changing at a constant rate.    This is shown by a straight line on a speed time graph.

A tachograph on a lorry is a speed time graph.  It is a very accurate speedometer (in kilometres per hour) which plots a speed time graph on a small circular piece of paper.  The instrument plots a trace which you can see on the card.  There are strict rules on the hours that drivers of buses and lorries may work.  The tachograph records when the driver is driving, is at work not driving, is taking breaks, and resting.  The tachograph is not new; instruments for recording speed have been used on railways for over a century.

When you use a ticker-tape timer, you are using a tachograph.  It is rather old-fashioned, but dead simple and low-tech.

Here is some ticker tape: The space between each dot is 1/50 second (0.02 s). Therefore 10 spaces give us 1/5 second (0.2 s)

We can work out the speed:

speed (cm/s) = distance (cm)

time (s)

In this case, speed = 12 cm ÷ 0.2 s = 60 cm/s

When we go on a journey, we tend to think of the average speed:

average speed (km/h) = total distance (km)

total time (h)

A lorry may take 1 hour to do 70 km, so its average speed is 70 km/h. However its tachograph will show that its speed might be 0 at some points (waiting for traffic lights), but doing 100 km/h down a dual carriageway road.

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Working Out Acceleration or Distance from a Speed-Time Graph (HT ONLY)

To work out the gradient of a graph:

• Draw a large rise and a large run:

• Gradient = rise/run

• Is the graph is curved, we draw a tangent.

 Question 11 What is the acceleration of the car whose motion is shown in the graph below?  To work out the distance from a speed time graph, you need to work out the area.  If the graph consists of rectangles and triangles, it's simple.  Work out the area of each shape and add them together.

If the graph is complicated, you can get a good idea of the area by counting the squares.  If more than half a square is under the graph, count it.  If less than half, ignore it.  Generally you get pretty close to the area.

(There is a mathematical process called calculus that allows us to work out the area under a graph without counting squares, but that is way beyond what you need to know.)

 Question 12 Use the information below in the question.  The train's motion is that described in the graph above. The maximum speed of the train is 25 m/s The time interval OX is 45 s The time interval XY is 45 s The time interval YC is 20 s          What is: a)   The acceleration between O and A b)   The acceleration between B and C c)   The distance covered while the train is at constant speed d)   The total distance e)   The average speed? If we take air resistance into account, we see that the acceleration gets less and less, as shown in the blue graph. The red graph shows a rocket going into space.  It mass gets less as fuel is used up.  As the force remains constant, the acceleration gets bigger.

 Question 13 Use the graph above to discuss this situation.  Also use what you have learned in this topic and Topic 1. The Ford Mondeo at the start of this topic accelerates from 0 - 100 km/h in 10.5 s.  It has a maximum speed of 210 km/h.  A student argues that the car could reach 200 km/h in 21 seconds and its maximum speed in 22 s, and draws a speed time graph to show the idea. Discuss whether the student is right. Summary Speed = distance ÷ time Velocity is speed in a certain direction; Acceleration = change in speed ÷ time; Motion can be represented in distance time or speed time graphs; Gradient of the distance time graph is speed; Gradient of the speed time graph is acceleration; Area under the speed time graph gives distance.