a body of mass m accelarates uniformly from the rest to velocity v1 in time t,find the expression for instantaneous power deliver to the body as the function of t?.......can any one help me with this......please.....

To find the expression for instantaneous power delivered to a body as a function of time, we need to understand a few concepts.

Power (P) is defined as the rate at which work is done or the rate of energy transfer. Mathematically, it can be expressed as:

P = ΔW / Δt

Where ΔW is the change in work done and Δt is the change in time.

Now, let's consider the given scenario. A body of mass m accelerates uniformly from rest to velocity v1 in time t.

First, let's find the work done on the body. Work (W) is defined as the product of the force (F) applied on an object and the distance (d) over which the force is applied. In this case, the distance is the displacement of the body.

Since the body is accelerating uniformly, we can use the equations of motion to determine the displacement (d) in terms of time (t) and the initial and final velocities (v0 and v1, respectively).

The equation for displacement in terms of initial and final velocities and time is:

d = (v1^2 - v0^2) / (2a)

Where v0 is the initial velocity, a is the acceleration, and v1 is the final velocity.

Given that the body starts from rest (v0 = 0) and accelerates uniformly, the expression for displacement simplifies to:

d = v1^2 / (2a)

Next, we need to find the acceleration (a). The acceleration is defined as the change in velocity divided by the change in time.

a = (v1 - v0) / t
a = v1 / t

Substituting this value of acceleration into the expression for displacement, we have:

d = v1^2 / (2 * (v1 / t))
d = v1 * t / 2

Now, we can substitute this expression for displacement (d) into the equation for work done (W). Since the body starts from rest, the initial velocity (v0) is 0, and the work done can be given by:

W = F * d
W = m * a * d

Substituting the expression for acceleration (a) and displacement (d), we get:

W = m * (v1 / t) * (v1 * t / 2)
W = m * v1^2 / 2

Finally, we have the expression for work done (W). To find the expression for instantaneous power (P), we need to consider the change in work done with respect to time:

P = dW / dt

Differentiating the expression for work done (W) with respect to time (t), we get:

dW/dt = (m * v1^2 / 2) / dt
dW/dt = (m * v1^2 / 2) / (dt / dt)
dW/dt = m * v1^2 / 2

Therefore, the expression for instantaneous power delivered to the body as a function of time is:

P = m * v1^2 / 2