Distance, displacement, speed, velocity and acceleration

Distance is the separation of two points in space. It is a scalar quantity since it has magnitude only. The SI unit of distance is the metre (m). Long distances may be measured in kilometres (km) while short distances may be measured in centimetres (cm) or millimetres (mm). It should be noted that in determining the distance between two points the direction at any point along the path is not considered. Direction may keep on changing (Fig. 1.1) or remain constant (Fig. 1.2).

Displacement is the shortest distance between two points in the direction of motion. Displacement is, therefore, along a straight line. An arrowhead in diagrams indicates the direction of motion (Fig. 1.3). Since displacement has both magnitude and direction, it is a vector quantity. Displacement is defined as the distance covered in a specific direction. The SI unit of displacement is the metre (m).

The speed of a body is defined as the distance covered by the body in unit time. Thus

The SI unit of speed is metres per second (m/s). Other units of speed such as kilometres per hour (km/h) and centimetres per second (cm/s) are also commonly used.

When a body covers equal distances in unit time intervals, it is said to move with uniform speed. However, quite often a body moving between two points does so at varying speeds. Such a body is said to move with non-uniform speed. In such a case the speed between the two points is called average speed.

The speed of a body in a specified direction is called velocity. Therefore:

velocity\mathrm{=\frac{\mathrm{\text{distance covered in a particular direction}}}{\mathrm{time\ taken}}}

or

\mathrm{velocity=\frac{\mathrm{displacement}}{time\ taken}}

Velocity is defined as the displacement covered in unit time i.e the rate of change of displacement. Velocity is a vector quantity. In some cases, the velocity of a moving body keeps on changing. In such cases, it is better to consider the average velocity of the body.

The SI unit of velocity is metres per second (m/s). The direction of velocity should always be indicated.

Velocity of a moving object can be determined in a number of ways. One of the methods is to use a ticker-tape timer.

The ticker-tape timer is an electrical vibrator which moves a metal pin up and down 50 times every second (Fig. 1.6 (a)). This means that the time taken for one complete vibration is 0.02 s. Each time the pin moves downwards it presses on a carbon paper disc and makes a dot on the paper tape which passes underneath the carbon. The tape is attached to a moving body e.g. a trolley as shown in Fig. 1.6 (b). Each successive pair of dots represents a time interval of 0.02 s. The distance between any two successive pair of dots is the distance the object has moved in 0.02 s. The tape, therefore, records the distance moved and the time taken by a moving body.

Consider the ticker-tape shown in Fig. 1.9. Measure the distances AB, BC, CD, DE, EF, FG, GH, HI. What can you say about these distances? (All these distances are equal. They are distances the tape moves in equal intervals of time). What is the distance PQ? (3.0 cm). What is the time taken by the tape to travel the distance PQ? (As there are 5 spaces between P and Q, the time taken by the tape to travel the distance PQ is 5 × 0.02 s = 0.10 s).

Calculate the velocity between points P and Q.

Since the dots are equally spaced this velocity is the same at all points on the tape. The body has uniform velocity.

When the velocity of a body changes the body is said to be accelerating.

The SI unit of acceleration is metres per square second or ms^-2. Acceleration is a vector quantity. If the acceleration of a body is 4 ms^-2, it means that its velocity is increasing by 4 m/s every second. When the velocity of a body decreases, it is said to be decelerating or retarding. Deceleration or retardation is a negative acceleration.

The velocity of a body at a specific moment in time is called the instantaneous velocity.

Determining the instantaneous velocity of a trolley

Consider the tape given in Fig. 1.12 produced in an experiment using a ticker-tape timer. The instantaneous velocity of the trolley at A and B may be calculated as follows:

Measure the distance between the dots A₁ and A₂, the dots preceding and after point A respectively. Divide the distance A₁A₂ by the time taken to travel that distance A₁A₂ to get the velocity at point A.

Fig. 1.12: Ticker-tape

Thus velocity at point A =

=\frac{\mathrm{\text{distance}\ A_1A_2}}{\text{time taken to travel distance}\ \mathrm{A_1A_2}}

=\frac{\mathrm{A_1}\mathrm{A_2}}{0.02\times2}=\frac{\mathrm{A_1A_2}}{0.04}

Similarly velocity at B =\frac{\mathrm{B_1B_2}}{0.04}

Determining constant acceleration of a trolley

The tape of Fig. 1.13 was obtained for a trolley moving with constant acceleration.

Fig. 1.13: Ticker-tape

Let the velocity of the trolley at points A and B be v_A and v_B respectively.

Thus\ v_{\mathrm{A}}=\frac{\mathrm{A_1A_2}}{0.04}and\ v_{\mathrm{B}}=\frac{\mathrm{B1_1B_2}}{0.04}

Acceleration between A and B =

=\frac{\mathrm{change\ in\ velocity\ from\ A\ to\ B}}{\mathrm{change\ in\ time\ from\ A\ to\ B}}

=\frac{v_{\mathrm{B}}-v_{\mathrm{A}}}{0.02\times4}=\frac{v\mathrm{_B}-v_{\mathrm{A}}}{0.08}