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Question

The acceleration of a particle starting from rest varies with time according to relation a=αt+β. What will be the velocity of the particle after time t?


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Solution

Acceleration:

  1. The rate of change of velocity is defined as acceleration.
  2. It is a vector quantity.

Step 1: Given data

The acceleration of the particle, a=αt+β

Step 2: Formula used:

We know,

The acceleration of a particle,

a=dvdt

Where dv is change in velocity and dt is change in time.

Step 3: Calculation

a=αt+βdvdt=αt+βdv=αt+βdt

On integrating,

0vdv=0t(αt+β)dtv0v=αt22+βt+C0tv=αt22+βt+C

where C is the constant of integration.

Since a particle starts from rest, its initial velocity is zero.

Therefore when t=0, v=0.

v=αt22+βt+CC=0

Therefore, the velocity of the particle after a time t will beαt22+βt.


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