Describing Motion Around Us

Class 09 Science

Everything in nature is in motion, from massive astronomical objects to subatomic particles. When an object moves in a straight line, its motion is called linear motion.

Motion in a Straight Line

Position of the object

The distance and direction of the object with respect to the reference point, at any instant of time, describes the position of the object at that instant of time.

If the position of the object with respect to the reference point changes with time, the object is said to be in motion. On the other hand, the object is said to be at rest if its position with respect to the reference point does not change with time.

For the object moving in a straight line, the object can move only in one of the two directions - forward and backward. Thus, the direction is represented by plus (+) and minus (-) signs. Positions to the right of the reference point O are generally taken as positive and to the left of O as negative.

Displacement

Displacement is the net change in the position of an object between the two given instants of time. The magnitude of displacement is the distance between the object’s positions at the two instants. The direction of displacement is specified from the position at the first instant towards the position at the second instant.

To describe the total distance travelled, only the numerical value (with units) is required, not the direction of motion. The SI unit for both is the metre (m).

Average Speed

The average speed of an object is the total distance travelled divided by the time interval during which this distance is covered.

$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Time Interval}} $$

Since distance travelled has no direction (but only a numerical value), the average speed, which is calculated from distance travelled, also has no direction but only a numerical value.

Uniform and Non-Uniform Motion

If an object moving in a straight line travels equal distances in equal intervals of time (for all possible choices of time intervals), it is said to be in uniform motion in a straight line. In this case, the object moves at a constant speed.

On the other hand, if the object travels unequal distances in equal intervals of time, then it is in non-uniform motion in a straight line. In this case, the object moves with increasing speed or decreasing speed, or a combination of both. If the distances travelled in the successive intervals of times are increasing, its speed is increasing.

Average Velocity

The speed tells us how fast an object is moving but it provides no information about the direction of motion. There are many situations where along with the speed, you also need to know the direction of motion to get a complete picture.

The average velocity of an object in a time interval is the change in the position (or displacement) divided by the time interval in which the change in position (or displacement) occurs.

$$ \text{Average velocity} = \frac{\text{Displacement}}{\text{Time Interval}} $$

To express the average velocity, you need to specify its magnitude as well as the direction. The direction of the velocity is the same as the direction of displacement and is indicated by a + or - sign.

The SI unit of average speed and average velocity are the same. It is metre per second which is represented by m s-1 or m/s. It is also commonly measured in kilometre per hour (km h-1).

Example: Sarang takes 50 seconds to swim from one end to the other end (25 m) and back in the swimming pool. Find his average speed and average velocity within the time interval of 50 s.

Total distance travelled in 50 s = 50 m

Displacement in 50 s = 0 m

$$ \text{average speed} = \frac{\text{total distance}}{\text{time}} = \frac{50}{50} = 1 \, \text{m/s} $$

$$ \text{average velocity} = \frac{\text{displacement}}{\text{time}} = \frac{0}{50} = 0 \, \text{m/s} $$

Average Acceleration

The average acceleration of an object over a time interval is the change in its velocity divided by the time interval.

$$ \text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time interval}} $$

$$ a = \frac{v - u}{t} $$

The SI unit of average acceleration is m s-2 or m/s2. Like displacement and velocity, you need to specify the magnitude as well as the direction of acceleration. For motion in a straight line, if the magnitude of velocity is increasing in a given time interval, the average acceleration is in the direction of velocity. Whereas, the average acceleration is opposite to the direction of velocity if the magnitude of velocity is decreasing.

The average acceleration can result from change in the magnitude of velocity or change in its direction, or both.

Example: A bus is moving on a long straight highway with a velocity of 36 km h-1. The driver presses the accelerator for a time interval of 10 s and velocity of the bus increases to 54 km h-1. For some time, the bus moves at a constant velocity. Then, the driver notices an obstacle on the road ahead and presses the brake. The bus comes to a stop in a time interval of 5 s. Find the average acceleration in the two time intervals, (i) when the accelerator was pressed, and (ii) when the brakes were pressed.

(i) When the driver presses the accelerator

  • u = 36 km/h = 10 m/s
  • v = 54 km/h = 15 m/s
  • t = 10 s
  • a = ?

$$ a = \frac{v - u}{t} $$

$$ a = \frac{15 - 10}{10} = 0.5 \, \text{m/s}^2 $$

Since the magnitude of velocity of the bus is increasing, the acceleration is acting in the direction of velocity.

(ii) When the driver presses the brake

  • u = 54 km/h = 15 m/s
  • v = 0 m/s
  • t = 5 s
  • a = ?

$$ a = \frac{0 - 15}{5} = -3 \, \text{m/s}^2 $$

The minus sign indicates that the acceleration is acting opposite to the direction of velocity (since the magnitude of velocity of the bus is decreasing).

Graphical Representation of Motion

Graphical representation provides a visual representation of how position, velocity and acceleration change with time. Such graphs help in comparing the motion of two objects, in calculating physical quantities, or in identifying whether the motion is uniform or non-uniform.

Position-time Graphs

The position-time graph represents the motion of object - the change in its position with time.

A straight line parallel to the time axis on a position-time graph represents a stationary object. A straight line position-time graph indicates that the object is moving with a constant velocity. On the other hand, a curved position-time graph indicates that the velocity is not constant, and thus, the object is in accelerated motion.

From the position-time graph, you can find the position of an object at each instant of time. Velocity can be determined from the slope of the straight line on position-time graph.

Velocity-time Graphs

The velocity-time graph of an object in motion represents the change in its velocity with time.

When the velocity is constant, the velocity-time graph is a straight line parallel to the x-axis and acceleration is zero. A straight line velocity-time graph indicates that the velocity is increasing or decreasing with a constant acceleration.

The slope of the line on velocity-time graph gives how fast the velocity is changing with time, i.e., the acceleration. The area enclosed by the velocity-time graph and the time axis for a desired time interval is equal to the displacement in that time interval.

Equations for Motion

For the motion of an object in a straight line with constant acceleration, the five physical quantities - displacement (s), time interval (t), initial velocity (u), final velocity (v) and acceleration (a), can be related by the following set of equations:

$$ v = u + at $$

$$ s = ut + \frac{1}{2}at^2 $$

$$ v^2 = u^2 + 2as $$

These are known as kinematic equations. These equations provide a mathematical description of how the motion of an object changes with time.

Example: Suppose a car is moving on a highway and brakes are applied, which cause an acceleration of -4 m s-2. How much will be the distance travelled by the car before coming to a stop, if the car was moving with a velocity of (i) 54 km h-1, and (ii) 108 km h-1 when the brakes were applied?

Given:

  • a = -4 m s-2
  • v = 0 m s-1

(i) u = 54 km/h = 15 m/s

(ii) u = 108 km/h = 30 m/s

$$ v^2 = u^2 + 2as $$

Substituting the value of v, u and a, we obtain (i) 28.1 m, and (ii) 112.5 m.

Uniform Circular Motion

When an object moves in a circular path, its motion is called circular motion.

The distance travelled in making one revolution (going round the circle once) is equal to the circumference of the circle. So, if the radius of the circular path is R, the distance travelled by the object in making one revolution is 2πR. On the other hand, the displacement is zero, since the object comes back to its original position after making one revolution.

If an object takes time T to make one revolution, its average speed will be

$$ v = \frac{2\pi R}{T} $$

When an object moves in a circular path with constant (uniform) speed, its motion is called uniform circular motion.

Acceleration is non-zero if the velocity of an object changes. The velocity changes if either its magnitude or direction, or both changes. In uniform circular motion, the motion of the object is accelerated because the direction of its velocity continuously changes.