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

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

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.

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.

In some cases, bearings are used for the direction part of displacements. The angle quoted in a

**bearing**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**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.

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.

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.

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**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. |

Acceleration is the change in velocity per second and its units are normally stated as

**ms**or sometimes as m/s/s. An acceleration of 1 ms

^{−2}^{−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:

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.

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.

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 change in velocity (an acceleration) is always due to the action of an

**unbalanced force**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.

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,

*t*

_{1}and

*t*

_{2}, along with the time taken to travel between the light gates,

*t*

_{3}.

*t*_{1}is the time for which the first light beam is interrupted.*t*_{2}is the time for which the second light beam is interrupted.*t*_{3}is the time taken to travel between the light beams.

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

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.

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.

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.

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