work done in time t on a body of mass m, which is accelerated uniformly from rest to a speed v in time t1 as a function of time t, is given by??

To determine the work done on a body of mass \( m \) that is accelerated uniformly from rest to a speed \( v \) in a time \( t_1 \), we can break down the problem using some fundamental concepts from physics, particularly kinematics and the work-energy principle.

Understanding Uniform Acceleration

When a body accelerates uniformly, it means that its velocity changes at a constant rate. The formula for acceleration \( a \) can be expressed as:

• a = (v - u) / t_1

Here, \( u \) is the initial velocity, which is 0 since the body starts from rest. Therefore, the acceleration simplifies to:

• a = v / t_1

Calculating the Distance Traveled

The distance \( s \) traveled by the body during this time can be calculated using the formula for distance under uniform acceleration:

• s = ut + (1/2)at^2

Substituting \( u = 0 \) and \( a = v / t_1 \), we get:

• s = (1/2)(v / t_1)t_1^2 = (1/2)vt_1

Work Done on the Body

The work done \( W \) on the body is equal to the change in kinetic energy as it accelerates from rest to speed \( v \). The kinetic energy \( KE \) at speed \( v \) is given by:

• KE = (1/2)mv^2

Since the initial kinetic energy is zero (the body starts from rest), the work done is simply:

• W = KE - 0 = (1/2)mv^2

Expressing Work as a Function of Time

To express the work done as a function of time \( t \), we need to relate \( v \) to \( t \). Since the body accelerates uniformly, the velocity at any time \( t \) (where \( 0 \leq t \leq t_1 \)) can be expressed as:

• v(t) = (v / t_1)t

Now, substituting this expression for \( v \) into the kinetic energy formula gives:

• KE(t) = (1/2)m(v(t))^2 = (1/2)m((v / t_1)t)^2 = (1/2)m(v^2 / t_1^2)t^2

Thus, the work done on the body as a function of time \( t \) is:

• W(t) = (1/2)m(v^2 / t_1^2)t^2

Summary

In summary, the work done on a body of mass \( m \) that is uniformly accelerated from rest to a speed \( v \) in time \( t_1 \) can be expressed as:

• W(t) = (1/2)m(v^2 / t_1^2)t^2

This equation shows how the work done varies with time during the acceleration phase, illustrating the relationship between force, mass, and motion in a clear and logical manner.