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Chapter 4: Problem 495

A body of mass \(\mathrm{m}\) is accelerated uniformly from rest to a speed \(\mathrm{v}\) in time \(\mathrm{T}\). The instantaneous power delivered to the body in terms of time is given by..... (A) $\left[\left\\{\mathrm{mv}^{2}\right\\} /\left\\{\mathrm{T}^{2}\right\\}\right] \cdot \mathrm{t}$ (B) $\left[\left\\{\mathrm{mv}^{2}\right\\} /\left\\{\mathrm{T}^{2}\right\\}\right] \cdot \mathrm{t}^{2}$ (C) $\left[\left\\{\mathrm{mv}^{2}\right\\} /\\{2 \mathrm{~T}\\}\right] \cdot \mathrm{t}$ (D) $\left[\left\\{\mathrm{mv}^{2}\right\\} /\left\\{2 \mathrm{~T}^{2}\right\\}\right] \cdot \mathrm{t}^{2}$

Short Answer

Expert verified
The instantaneous power delivered to the body in terms of time is given by \(P(t) = \frac{mv^2}{T^2}t\).

Step by step solution

01

1. Find the acceleration of the object from given data

Use the kinematic equation for uniformly accelerated motion to find the acceleration: \(v = u + at\), where v is the final velocity, u is the initial velocity (0 since the object starts from rest), a is the acceleration, and t is the time. Since we are given that the object accelerates for time T, we can substitute the values: \(v = 0 + aT\) Now solve for acceleration: \(a = v/T\)
02

2. Calculate the force acting on the object using Newton's second law

Use Newton's second law of motion to find the force acting on the object: \(F = ma\) Substitute the value of found acceleration in the previous step: \(F = m(v/T)\) Simplify the equation: \(F = mv/T\)
03

3. Calculate the instantaneous power of the object

Use the formula for power to find the instantaneous power P in terms of time t: \(P(t) = Fv(t)\) Since we know that the object starts from rest and accelerates uniformly, we can find the velocity v(t) as a function of time using the kinematic equation: \(v(t) = u + at\), where u = 0. Substitute the value of acceleration and u in the equation: \(v(t) = (v/T)t\) Now, substitute the expressions for F and v(t) from the previous steps: \(P(t) = (mv/T)(v/T)t\) Finally, simplify the equation to get the expression for the instantaneous power in terms of time: \(P(t) = \frac{mv^2}{T^2}t\) This corresponds to option (A).

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