Introduction to Accelerated Motion:

When driving a vehicle, its velocity increases and decreases randomly. This changing velocity is referred to as accelerated motion. In contrast, a body travels in uniform motion if it moves with constant velocity, covering equal displacements in equal intervals of time. If the velocity's magnitude or direction changes, the body is not in uniform motion. For a body in uniform motion, the net external force acting on it is zero, and the reverse is also true.

Uniform Acceleration:

Uniform acceleration occurs when a body is subjected to a constant external force, resulting in a constant acceleration. The direction of this acceleration plays a crucial role in the change of velocity, which may not always align with the direction of motion. For instance, if a car travels at a constant speed of 60 km/hour, covering a distance of 1 km/minute, it is an example of uniform acceleration.

Kinematic Equations of Motion:

Consider an object starting with an initial velocity “u”. After time “t”, its velocity changes to “v “, with a uniform acceleration “a” and distance travelled “s”. The following kinematic equations of uniformly accelerated motion are derived:

First Equation of Motion:

Given:

u: Initial velocity

v: Final velocity

a: Uniform acceleration

t: Time

From the graph:

Slope= Acceleration(a)= change in velocity time

Change in velocity = AB= v⃗ −u⃗

Time = AD = t

a = v⃗ −ut

Solving this we get the first equation of motion:

v⃗ =u⃗ +a⃗ t

Second Equation of Motion:

Considering the same object with initial velocity “u” and uniform acceleration “a”:

- Distance covered in time t: s

Displacement (s) = Area of ABD + Area of ADOE

1 × AB× AD+ AE ×OE, AB= dc= at

= 1 at× t+ ut

Hence, the second equation of motion is:

s⃗ =u⃗ t+1/2 a⃗ t²

Velocity-Time Graph of Uniformly Accelerated Motion:

The velocity-time graph for uniformly accelerated motion is a straight line inclined towards the time axis. If the acceleration is positive and constant, the graph slopes upward. Conversely, for negative constant acceleration, the graph slopes downward.

Some Solved Examples for Students:

Example 1:

An automobile travelling at 60 km/h can stop within 20 m after applying the brakes. If the car's speed doubles to 120 km/h, the stopping distance will be:

A. 60m

B. 40m

C. 20m

D. 80m

Answer:

D. 80m

Example 2:

A cart moving in the positive direction slows down under a constant backward pull until it momentarily stops and reverses direction. What are the signs of the cart’s velocity and acceleration at the moment it stops?

A. Velocity: 0, Acceleration: +

B. Acceleration: 0, Velocity: +

C. Velocity: 0, Acceleration: –

Answer:

C. Velocity: 0, Acceleration: –

The cart's velocity is zero at the moment it stops, and since it is about to accelerate in the negative direction to change its velocity from zero to a negative value, the acceleration is negative.