Velocity and Acceleration
Introduction

Knowing the size of a distance or speed is often useful for making comparisons. For example, for some purposes it may be sufficient to know that Inverness is further from London than it is from Birmingham. However, in many everyday situations it is not enough to know the speed or distance. Suppose you want to get to a destination 200 km away at a speed of 50 kmh−1. That information is enough to tell you that the journey will take four hours. But what are the chances of arriving safely at your destination if you set off in a random direction?

Specifying a destination will involve stating a direction as well as a distance travelled. Quantities where direction is equally as important as magnitude are called vector quantities. Non-vector quantities such as mass and time are called scalars.

In the situations met so far, direction has not been a major consideration. However, when a ship is lost at sea, locating the survivors depends upon knowing their exact location. In this unit you will be introduced to some new terms whose definitions are more specific than the ideas of distance and speed that you have previously met.

Displacement and velocity
In a maritime rescue, the rescue services need to know how far offshore the ship sank and in what direction they must travel to get there. The distance and direction parts of the location are both equally important. In physics the term displacement is used to describe a case where the position of an object is specified in terms of both its distance and direction from a reference point.

Which type of quantity is
displacement
An object's displacement quotes both its bearing and distances relative to a fixed reference point.
displacement
?

In some cases, bearings are used for the direction part of displacements. The angle quoted in a
bearing
Bearings are a method of stating direction by measuring the angle, clockwise, from due north.
bearing
is measured clockwise from north (000).

 Figure 1. Bearings are used in position location.
Match each of the points in Fig.2 with the corresponding displacement.
•  Point A 10 km, bearing 120 13 km, bearing 045 30 km, bearing 240 8 km, bearing 330 Point B 10 km, bearing 120 13 km, bearing 045 30 km, bearing 240 8 km, bearing 330 Point C 10 km, bearing 120 13 km, bearing 045 30 km, bearing 240 8 km, bearing 330 Point D 10 km, bearing 120 13 km, bearing 045 30 km, bearing 240 8 km, bearing 330
 Figure 2. Map.

The definition of average speed takes no account of the direction in which the object is moving.

When defining the velocity with which an object is moving, we do need to include the direction.
Velocity
An object's velocity states both the speed and direction of motion relative to a fixed reference point.
Velocity
is defined as

Velocity can be stated in words in a number of different ways.

 The velocity of a moving object is the change in its displacement per second.
 Changes per second are sometimes called rates of change.
 So the velocity of an object can also be described as its rate of change of displacement.

Study the situations in Fig.3 and answer the following questions. All balls are travelling with the same speeds.

Click on the figure below to interact with the model.

 Figure 3.  Bouncing balls.

Decide which phrase correctly describes the velocities of the balls in each box.
•  The velocities of the balls in Box 1 Always the same Never the same Sometimes the same The velocities of the balls in Box 2 Always the same Never the same Sometimes the same The velocities of the balls in Box 3 Always the same Never the same Sometimes the same

Velocity values, stating both a speed and a direction of travel, can be used to locate the destination reached when travelling for a certain time.

Move the line in Fig.4 by dragging on the green circle. Then match each of the destinations with the appropriate journey details.

• Reaching point A from O in 45 minutes involves travelling at a speed of  kmh−1 on a bearing of .

Reaching point B from O in 4 hours involves travelling at a speed of kmh−1 on a bearing of .

Reaching point C from O in 8 hours involves travelling at a speed of kmh−1 on a bearing of .

Reaching point D from O in 10 minutes involves travelling at a speed of  kmh−1 on a bearing of .
 Figure 4. Journey directions, distances and times.

In many examples we are only required to consider motion in a straight line, so there are only two possible directions. These may be to the left and to the right. In other situations it might be more helpful to use up and down or north and south.

To simplify the direction part of a stated velocity in such situations we can use the signs + and − to distinguish opposite directions of travel. The choice of which to make positive is purely arbitrary, but needs to be made at the start of an example or question and must be applied consistently throughout.

If a speed of 20 ms−1 in a northerly direction is written as +20 ms−1, then what is the velocity of an object moving due south at a speed of 5 ms−1?
If a speed of 4 ms−1 to the left is written as −4 ms−1, then what is the velocity of an object travelling 10 m to the right of an observation point in 4 seconds?
Acceleration
Motor car manufacturers, keen to show how quickly their latest sports car can speed up, sometimes publish the time the car takes to go from 0 to 60 mph. Other manufacturers will quote the 0 to 100 mph figure, so direct comparison can be difficult. Acceleration is a more scientific way of describing how quickly a moving object can change its velocity. It is defined as follows.

Acceleration
An object's acceleration is its rate of change of velocity.
Acceleration
can be stated in words in a number of different ways.

 The acceleration of a moving object is the change in its velocity per second.
 Changes per second are sometimes called rates of change.
 So the acceleration of an object can also be described as its rate of change of velocity.

A car accelerates from rest to 27.8 ms−1 in 5 seconds. What is the acceleration of the car?

Acceleration is the change in velocity per second and its units are normally stated as ms−2 or sometimes as m/s/s. An acceleration of 1 ms−2 means that an object's speed is increasing by 1 ms−1 every second.

Even when the performance of different cars is stated in different ways we can make direct comparison by calculating their accelerations using the formula:

 Figure 5. Comparing typical car performances.

The force of gravity acting on an object results in an acceleration of 9.81 ms−2, although this is sometimes approximated to 10 ms−2. This means that every second, a falling object travels approximately 10 m/s faster than in the previous second. Three seconds after being dropped from the top of a tall building an object will be travelling at an approximate speed of 30 ms−1. This is roughly the same speed as a fast-moving car.

A ball dropped from rest from the top of a tall building takes 2.65 s to hit the ground. What is its approximate velocity on impact?
Retardation
In the examples studied so far the change in velocity has been caused by an object speeding up. Velocity changes will also happen when an object slows down. This is frequently referred to as a deceleration or retardation and, in mathematical examples, it is indicated as a negative acceleration. An object projected vertically upwards will slow down as it rises towards its highest point. As it rises, its speed reduces by about 10 ms−1 every second. The acceleration in this case is stated as −10 ms−2.

Drag the ball in Fig.6 to the top of its range of motion and release it. As it falls its velocity at different times is recorded on a graph.

Use the pause button in the experiment and the velocity versus time graph to answer the questions below.

Click on the figure below to interact with the model.

 Figure 6.  Bouncing ball.

What is the balls velocity at the instant it is released?
What is the ball's approximate velocity one second after it is released?
Complete the following statements.

• As the ball in Fig.6 rises, its speed . When rising it is .When the ball is falling its speed , so it is

The velocity–time graph in Fig.6 shows times when the ball is either speeding up or slowing down. Its velocity is zero twice during each bounce: once at its highest point and once when in contact with the ground.

Accelerating at constant speed
Earlier we stated that velocity is a
vector
A vector quantity is specified fully only when its magnitude and direction are both quoted.
vector
quantity. This means that the velocity of a moving object will change when the direction of motion changes, even if the speed stays the same! Consequently the ball in Fig.7 is accelerating even though its speed around the perimeter of the circle is not altering.

 Figure 7. Ball moving at a constant speed but accelerating.

A change in velocity (an acceleration) is always due to the action of an
unbalanced force
Unbalanced forces occur in situations where the force acting in one direction is greater than the force acting in the opposite direction.
unbalanced force
. In Fig.7 the force in the string keeps the ball moving in a circle. Press the button to snap the string. You will see that the ball flies off. You will learn more about this situation when studying circular motion.

Measuring acceleration
To measure the acceleration of a car, you need to determine a change in velocity and the time taken for this change to occur. To find the change in velocity you need to find an initial velocity and a final velocity. In mathematical notation, the final velocity, initial velocity, and the time for the change to occur are given the symbols v, u, and t. So the acceleration can be defined as:

Measuring two velocities in the lab can be achieved using the type of apparatus shown in Fig.8.

 Figure 8. Determining acceleration.

The interrupt card on top of the trolley is 0.1 m long, so the velocity at each light can be found from:

The timing system connected to the light gates must be capable of measuring the times for which each light gate is interrupted, t1 and t2, along with the time taken to travel between the light gates, t3.

• t1 is the time for which the first light beam is interrupted.

• t2 is the time for which the second light beam is interrupted.

• t3 is the time taken to travel between the light beams.

In one run of the experiment in Fig.8 the card is 10 cm long and the interrupt time t1 = 0.058 s. Complete the following statements (to 2 d.p.).

• The average speed through the first light gate is  ms−1.

The interrupt time for the second light gate is 0.040 s, so the average speed through the second light gate is  ms−1.

The time taken to travel between the two light gates is 0.34 s, so the acceleration is  ms−2.

A simpler way of measuring the acceleration uses a single light gate and a double interrupt card as shown in Fig.9.

 Figure 9. Determining acceleration with a single card.

Careful thought will show that when this card passes through the light gate it produces the three time measurements needed to calculate the initial velocity, the final velocity, and finally the acceleration.

In one run of the experiment in Fig.9 the sections of card that interrupt the light beam are each 6 cm long and the interrupt time t1 = 0.019 s. Complete the following statements (2 d.p.).

• The average speed during the first interrupt is  ms−1.

The interrupt time for the second light gate is 0.016 s, so the average speed through the second light gate is  ms−1.

The time between the two interrupts is 0.087 s, so the acceleration is  ms−2.
Summary

The location of one point relative to another is specified by the distance between the points and the direction of one from the other.

Stating a point's displacement involves giving its distance from a reference point and the direction relative to the reference point.

Velocity is defined as the rate of change of displacement. Mathematically,

Changes in either speed or direction will cause a change in an object's velocity.

Acceleration is defined as the rate of change of velocity. Mathematically,

Accelerations are caused by forces which change either the speed or direction of a moving object.

Accelerations are measured in the laboratory with light gates and timers.

Exercises
1. A robot in the production line at a car factory moves north (000) for 1.2 m. It turns right and travels a further 1.6 m. Finally, it turns right again and travels another 1.2 m.

• How far has the robot travelled during this operation?
m   (to the nearest whole number)

What is the robot's final displacement from its starting position?
Distance:  m   (to 1 d.p.);   Bearing:
2. An unpowered glider is towed at constant altitude by a light plane which travels due north (000) for a distance of 500 m. After being released from the plane the glider flies at the same height in a northeasterly direction (045) for a further 3 km. What is the glider's displacement from its starting position?

• Distance:  km   (to 2 d.p.);   Bearing:
3. A student making notes about velocity writes down the following statements. Decide whether they are true or false.
•  Velocity is a vector quantity. False True To state a velocity value properly you must indicate the direction. False True Velocity is the rate of change of displacement. False True Velocity is another name for speed. False True
4. A light aircraft flying at 50 ms−1 due north (bearing 000) is blown off course by a wind with a velocity of 30 ms−1 from the south-west (bearing 225).

• What is the velocity of the plane as seen by an observer on the ground?
Speed:  ms−1   (to 2 d.p.); Bearing:

At a particular instant the plane is directly over a marker. How far away from this position is the plane 5 minutes later?
km   (to 1 d.p.)
 Figure 10. An aircraft flying due north.
5. To relieve the boredom during a long flight, a golfer takes his putter and putts a golf ball 10 m along the aisle towards the front of the plane and into a practice hole.

• The ball takes 10 seconds to travel along the aisle and the plane is flying at 500 ms−1. What is the speed of the ball relative to the ground?
ms−1   (to the nearest whole number)

How far, relative to the ground, does the ball travel?
m   (to the nearest whole number)
6. A student making notes on acceleration writes down the following statements. Are they true or false?
•  Acceleration is a vector quantity. False True Acceleration is defined as the change in velocity in unit time. False True The gradient of a velocity–time graph gives acceleration. False True If we are told that an object has accelerated, we know that its speed must have changed. False True
7. Car A can accelerate from 3 ms−1 to 24 ms−1 in 6 seconds. Car B is quoted as being able to accelerate from rest to 30 ms−1 in 9 seconds.

• Calculate the acceleration for car A.
ms−2   (to 1 d.p.)

Calculate the acceleration for car B.
ms−2   (to 1 d.p.)
8. A cyclist, travelling north along a straight level road, reaches a maximum speed of 12 ms−1 by accelerating at 2.5 ms−2 for 3 seconds. What was the velocity before the acceleration started?
•  ms−1   (to 1 d.p.)
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