Problem 59 - Master Resource Book in JEE Main Physics

Question

A body of mass \(m\) is accelerated uniformly for 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 2005] (a) \(\frac{1}{2} \frac{m v^{2}}{T^{2}} t^{2}\) (b) \(\frac{1}{2} \frac{m v^{2}}{T^{2}} t\) (c) \(\frac{m v^{2}}{T^{2}} t^{2}\) (d) \(\frac{m v^{2}}{T^{2}} t\)

Step-by-Step Solution

Answer

The correct answer is (d) \(\frac{m v^2}{T^2} t\).

Step 1: Understand the Power Formula

The power delivered to a body is the rate at which work is done or energy is transferred. For a body accelerating uniformly, power at any time \(t\) can be expressed as \(P(t) = F \cdot v(t)\), where \(F\) is the force applied to the body and \(v(t)\) is the velocity at time \(t\).

Step 2: Calculate Force

Since the body is uniformly accelerated from rest to speed \(v\) in time \(T\), the acceleration \(a\) is \(\frac{v}{T}\). Using Newton's second law, \(F = m \cdot a\). Therefore, \(F = m \cdot \frac{v}{T}\).

Step 3: Determine Velocity as a Function of Time

For uniform acceleration from rest, the velocity \(v(t)\) at time \(t\) is given by \(v(t) = a \cdot t = \frac{v}{T} \cdot t\).

Step 4: Substitute into Power Formula

Substitute \(F = m \cdot \frac{v}{T}\) and \(v(t) = \frac{v}{T} \cdot t\) into the power formula: \(P(t) = (m \cdot \frac{v}{T}) \cdot (\frac{v}{T} \cdot t) = \frac{m v^2}{T^2} t\).

Step 5: Compare with Given Options

The derived expression for power \(P(t) = \frac{m v^2}{T^2} t\) matches option (d).

Key Concepts

Understanding Uniform Acceleration

Uniform acceleration occurs when an object speeds up or slows down at a constant rate. This means the acceleration does not change over time. When we say a body goes from rest, it starts with zero velocity. If it's uniformly accelerated, it gains speed steadily over a period of time.
To find out how acceleration affects velocity over time, we use the formula:

• Acceleration (\(a\)) = Change in Velocity / Change in Time
• In this scenario, acceleration is \(a = \frac{v}{T}\).

Here, \(v\) is the final speed reached at time \(T\).Using this simple formula, we can express how the velocity of a body changes during uniform acceleration.

Applying Newton's Second Law

Newton's Second Law of Motion is a cornerstone of classical physics. It establishes a clear link between force, mass, and acceleration. The law states that the force acting on a body is equal to the mass of the body times its acceleration: \(F = m imes a\).
In our problem, since the acceleration is uniform, we know it as a constant \(\frac{v}{T}\). So, applying Newton's Second Law, the force becomes:

• \(F = m \times \frac{v}{T}\)

This derived force formula helps us understand the power being exerted on the body. Power is derived later by connecting force with velocity.

Analyzing Velocity-Time Relationship

The velocity-time relationship is crucial for understanding motion under uniform acceleration. Here, the velocity at any time \(t\) from rest is obtained as follows:

• Given \(v(t) = a \times t\)
• Substitute the uniform acceleration \(a = \frac{v}{T}\) to get \(v(t) = \frac{v}{T} \times t\)

Analyzing this relationship allows us to delve into how the speed varies over time under constant acceleration.
Finally, substituting \(v(t)\) and the force \(F\) into the power formula gives us a time-dependent expression for power as the solution calculated: \(P(t) = \frac{m v^2}{T^2} t\). This equation describes how power progressively changes as the object accelerates from rest to speed \(v\).