A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time ‘t’ is proportional to

a) t^3/4

b) t^3/2

c) t^1/4

d) t^1/2

When examining the motion of a body being moved by a machine that delivers a constant power, we can derive the relationship between the distance moved and the time taken. Let's break this down step by step to find out how distance relates to time in this scenario.

The Basics of Power and Motion

Power is defined as the rate at which work is done or energy is transferred. When a machine exerts constant power \( P \), the work done on the body can be expressed as:

• Power (P) = Work done (W) / Time (t)
• Work done (W) = Force (F) × Distance (d)

Establishing Relationships

Since power is constant, we can rearrange the formula for power to find work done over a period:

W = P × t

Substituting for work done, we get:

F × d = P × t

Force and Acceleration

According to Newton's second law, force can also be expressed in terms of mass \( m \) and acceleration \( a \):

F = m × a

Now, if we consider that acceleration \( a \) can be defined as the change in velocity over time, and given that the body starts from rest, we can relate acceleration to distance. Using the kinematic equation:

d = (1/2) a t²

Relating Acceleration to Power

We can substitute \( a \) from our earlier force equation into the distance equation:

d = (1/2) (P / m) t²

From this, we see that distance \( d \) is proportional to the square of time \( t \) while factoring in the constants:

d ∝ t²

Finding the Correct Proportionality

However, we need to consider the relationship between distance and time when power is constant. The body accelerates as it moves, which affects the relationship. By integrating the power with respect to distance, we can derive a different proportionality:

From the earlier equations, we can derive that:

d ∝ t^(3/2)

This means that the distance moved by the body over time when powered with constant energy follows a cubic root relationship with respect to time. Thus, the correct answer to your question is:

b) t^(3/2)

A Quick Recap

• Power is constant, leading to a linear relationship with work done over time.
• As the body accelerates, the relationship between distance and time becomes a function of \( t^{3/2} \).
• This highlights how motion under constant power behaves differently from motion under constant force.

Understanding these concepts helps us grasp the dynamics of motion better, especially in systems where power input remains unchanged. If you have any more questions or need clarification on any aspect, feel free to ask!