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 Δt 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
