Question

When a body accelerates by beta t, what is the velocity after time t, when it starts from rest?​

Answer 1

Concept:

The rate at which an object's velocity with respect to time changes is called acceleration in mechanics. It is formulated as, a = dv/dt

Given:

acceleration = βt

time = t

Find:

We need to determine the velocity, v after time t, when the body starts from rest.

Solution:

The rate at which an object's velocity with respect to time changes is referred to as acceleration in mechanics. They are vector quantities. The direction of the net force acting on an object determines the direction of its acceleration.

Mathematically, acceleration is expressed as- a = dv/dt

Therefore, dv = adt

Taking integration on both sides as v = 0 to v and t = 0 to t

$\int\limits^v_0 dv$ =  $\int\limits^t_0 adt$

we have a = βt

Therefore,

$\int\limits^t_0 adt$ = $\int\limits^t_0$βtdt

[v]₀^v = β [t²/2]₀^t

v - 0 = β [t²/2 - 0]

Therefore, v = βt²/2

Thus, the velocity after time t, when a body starts from rest is βt²/2.

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