Question

A constant force acts on a body at rest, resulting it to move with constant acceleration. If power relates to time as \( P \propto t^n \), what is the value of \( n \)?

Hint:

When an object moves with constant acceleration, the velocity increases linearly with time, resulting in power being directly proportional to time.

Answer 1

Step 1: Write the expression for power.
The power \( P \) delivered by a force is given by the formula: \[ P = F \cdot v \] where \( F \) is the force and \( v \) is the velocity of the body. Since the force is constant, the acceleration \( a \) of the body is constant. The velocity \( v \) of the body under constant acceleration is given by: \[ v = u + at \] where \( u \) is the initial velocity (which is 0, as the body starts from rest), \( a \) is the acceleration, and \( t \) is the time. Thus, the velocity can be written as: \[ v = at \]
Step 2: Express power in terms of time.
Substitute \( v = at \) into the expression for power: \[ P = F \cdot (at) \] Since \( F = ma \) (from \( F = ma \), where \( m \) is the mass), we get: \[ P = ma \cdot (at) = ma^2 t \]
Step 3: Relate power to time.
From the above equation, we see that the power is proportional to \( t \): \[ P \propto t \] Thus, comparing this with \( P \propto t^n \), we find that: \[ n = 1 \] Thus, the value of \( n \) is: \[ \boxed{1} \]