**IIT JEE Motion in a Straight Line with Acceleration | Acceleration: Motion in a Straight Line**

Velocity of a body is defined as the time rate of displacement, where as acceleration is defined as the time rate of change of velocity. Acceleration is a vector quantity. The motion may be uniformly accelerated motion or it may be non-uniformly accelerated, depending on how the velocity changes with time.

**Uniform Acceleration **

The acceleration of a body is said to be uniform if its velocity changes by equal amounts in equal intervals.

**Non-Uniform Acceleration**

The acceleration of a body is said to be non-uniform if its velocity changes by unequal amounts in equal intervals of time.

**Variable Acceleration**

The acceleration at any instant is obtained from the average acceleration by shrinking the time interval closer zero. As Î”t tends to zero average acceleration approaching a limiting value, which is the acceleration at that instant called instantaneous acceleration which is vector quantity.

Hence instantaneous acceleration of a particle at any instant is the rate at which its velocity is changing at that instant. Instantaneous acceleration at any point is the slope of the curve v (t) at that point as shown in figure above.

**Equations of Motion**

The relationship among different parameter like displacement velocity, acceleration can be derived using the concept of average acceleration and concept of average acceleration and instantaneous acceleration.

When acceleration is constant, a distinction between average acceleration and instantaneous acceleration loses its meaning,

**Refer this Simulation for Motion in a Straight Line**

**Illustration:**

The nucleus of helium atom (alpha-particle) travels inside a straight hollow tube of length 2.0 meters long which forms part of a particle accelerator. (a) If one assumes uniform acceleration, how long is the particle in the tube if it enters at a speed of 1000 meter/sec and leaves at 9000 meter/sec? (b) What is its acceleration during this interval?

**Solution:**

(a) We choose x-axis parallel to the tube, its positive direction being that in which the particle is moving and its origin at the tube entrance. We are given x and vx and we seek t. The acceleration ax is not involved. Hence we use equation 3, x = x0 + <v> t.

- The dispalcement remains unaffected due to shifting of origin from one point to the other.
- The displacement can have positive, negative or zero value.
- The dispalcement is never greater than the actual distance travelled.
- The displacement has unit of length.
- Velocity can be considered to be a combination of speed and direction.
- A change in either speed or direction of motion results in a change in velocity.
- It is not possible for a particle to possess zero speed with a non-zero velocity.
- A particle which completes one revolution, along a circular path, with uniform speed is said to possesss zero velocity and non-zero speed.
- In case a body moves with uniform velocity, along a straight line, its average speed is equal to its instantaneous speed.

**Question 1:-**

If a car at rest accelerates uniformly to a speed of 144 km/h in 20 s, it covers a distance of:

(a) 20 m (b) 800 m

**(c) 400 m ** (d) 1200 m

**Question 2:-**

If a body of mass 3 kg is dropped downwards, after 1 sec another ball is dropped downwards from the same point. What is the distance between them after 3 sec.

**(a) 25 m ** (b) 20 m

(c) 50 m (d) 9.8 m

**Question 3:-**

Two bodies of different masses ma and mb are dropped from two different heights a and b. The ratio of the times taken by the two to drop through these distances is:

(a) a:b (b) b:a

(c) a2:b2 ** (d) √a:√b**

** Question 4:-**

A ball takes t second to fall from a height h1 and 2t seconds to fall from a height h2. Then h1/h2 is:

(a) 0.5 ** (b) 0.25**

(c) 2 (d) 4