A body is initially at rest. It undergoes one-dimentional motion with constant
acceleration. The power delivered at time it at time 't' is proportional to
(a) t^1/2
(b) t
(c) t^3/2
(d) t^2

Ans. (b)

Question 6.10

A body is moving unidirectionally under the influence of a source of constant
power. Its displacement in time 't' is proportional to
(a) t^1/2
(b) t
(c) t^3/2
(d) t^2

Ans. (c)

I am unable to solve these questions. If anybody can solve it then please post
the solution here.

Let's break down these two questions step by step, focusing on the concepts of motion, acceleration, and power. Understanding the relationships between these variables will help clarify why the answers are as they are.

Question 1: Power Delivered with Constant Acceleration

The first question involves a body that starts from rest and moves with constant acceleration. We need to determine how power delivered at time 't' relates to time.

Understanding Power in Motion

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

• P = F * v

Where:

• F is the force applied on the body.
• v is the velocity of the body.

Finding Velocity and Force

Since the body is accelerating, we can use the equations of motion. The velocity (v) of a body under constant acceleration (a) after time 't' can be given by:

• v = u + at

Here, 'u' is the initial velocity, which is 0 (since the body starts from rest), so:

• v = at

The force (F) acting on the body can be expressed using Newton's second law:

• F = ma

Substituting the expression for force and velocity into the power equation gives us:

• P = (ma)(at) = ma^2t

Proportionality to Time

From the equation P = ma^2t, we can see that power is directly proportional to time (t). Therefore, the correct answer for the first question is:

• (b) t

Question 2: Displacement Under Constant Power

The second question involves a body moving under the influence of a constant power source. We need to find how displacement (s) relates to time (t).

Power and Displacement Relationship

When power is constant, we can express it as:

• P = F * v

Since power is constant, we can also relate it to the work done over time:

• P = W/t

Work done (W) is equal to the change in kinetic energy, which can be expressed as:

• W = (1/2)mv^2

Velocity in Terms of Power

From the constant power equation, we can rearrange it to find velocity:

• v = P/F

As the body moves, the displacement (s) can be calculated using the average velocity over time. Since power is constant, the velocity increases linearly with time:

• s = (1/2)vt = (1/2)(Pt/F)t

As we can see, displacement is proportional to:

• s ∝ t^3/2

Final Answer for Displacement

Thus, the displacement of the body under constant power is proportional to:

• (c) t^3/2

In summary, for the first question, power delivered at time 't' is proportional to time (t), and for the second question, displacement under constant power is proportional to time raised to the power of 3/2 (t^3/2). Understanding these relationships helps clarify how motion and power interact in physics.