Question

If a body at rest undergoes one dimensional motion with constant acceleration, then the power delivered to it at a time \( t \) is proportional to:

• A. \( \sqrt{t} \)
• B. \( t^2 \)
• C. \( t^3 \)
• D. \( t^{3/2} \)
• E. \( t \)

Hint:

The power delivered to an object under constant acceleration is proportional to the time \( t \), because both velocity and force are linearly related to time in such motion.

Answer 1

Answer: (E)

The equation for the motion of the body under constant acceleration is: \[ v = u + at \] where: - \( v \) is the velocity at time \( t \),
- \( u \) is the initial velocity (which is zero since the body is at rest),
- \( a \) is the constant acceleration,
- \( t \) is the time.
Therefore, the velocity at time \( t \) is: \[ v = at \] The power delivered to the body is the rate at which work is done, and it is given by: \[ P = Fv \] where: - \( P \) is the power,
- \( F \) is the force acting on the body,
- \( v \) is the velocity of the body.
The force \( F \) can be calculated using Newton’s second law: \[ F = ma \] where: - \( m \) is the mass of the body,
- \( a \) is the acceleration.
Thus, the power delivered to the body is: \[ P = ma \cdot v \] Substituting \( v = at \) into the equation: \[ P = ma \cdot (at) = ma^2t \] Thus, the power is directly proportional to time \( t \). \[ \boxed{t} \]