A body of mass m is accelerated uniformly from rest to a...

A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T$. The instantaneous power delivered to the body as a function of time is given by [AIEEE 2008] (a) $\frac{1}{2} \frac{m v^{2}}{T^{2}} t^{2}$ (b) $\frac{1}{2} \frac{m v^{2}}{T^{2}} t$ (c) $\frac{m v^{2}}{\tau^{2}} t^{2}$ (d) $\frac{m v^{2}}{T^{2}} t$

A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T$. The instantaneous power delivered to the body as a function of time is given by [AIEEE 2008]
(a) $\frac{1}{2} \frac{m v^{2}}{T^{2}} t^{2}$
(b) $\frac{1}{2} \frac{m v^{2}}{T^{2}} t$
(c) $\frac{m v^{2}}{\tau^{2}} t^{2}$
(d) $\frac{m v^{2}}{T^{2}} t$

Key Concepts

Kinematic Equations

Kinematic equations are a set of equations used to describe the motion of objects under constant acceleration. They help relate variables such as initial velocity, displacement, acceleration, and time. In problems involving uniform acceleration, these equations provide the necessary framework to derive expressions for velocity and displacement, which in turn facilitate the calculation of kinetic energy and power.

Newton's Second Law

Newton's Second Law, which states that force is equal to mass times acceleration (F = ma), is fundamental in connecting the study of motion to the dynamics of force and energy. This law underpins the derivation of expressions for instantaneous power in contexts involving uniform acceleration by linking the applied force with the resulting acceleration and the object's mass.

Instantaneous Power

Instantaneous power is defined as the rate at which work is done at a specific moment in time. It is mathematically expressed either as the derivative of work with respect to time or directly as the product of force and instantaneous velocity. This concept is central to understanding energy transfer in dynamics, as it quantifies how quickly work is performed in any mechanical process.

Uniform Acceleration

Uniform acceleration refers to motion in which the acceleration remains constant over time. In such cases, the relationship between velocity and time is linear, meaning that the velocity increases (or decreases) at a steady rate. This concept is crucial in kinematics and dynamics because it allows the derivation of other motion parameters, such as displacement and the time-dependent behavior of forces.

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