Question

If the displacement of a body varies as the square of elapsed time, then its

• A. velocity is constant
• B. velocity varies non-uniformly
• C. acceleration is constant
• D. acceleration changes continuously
• E. momentum is constant

Hint:

If \(s \propto t^n\), then acceleration is constant only when \(n = 2\).

Answer 1

The Correct Option is C

Concept: The nature of motion can be determined from the functional relationship between displacement and time. If displacement \(s\) is given as a function of time \(t\), then: \[ v = \frac{ds}{dt}, \quad a = \frac{dv}{dt} = \frac{d^2 s}{dt^2} \] Thus, by differentiating displacement with respect to time, we can obtain velocity and acceleration.

Step 1: Form the mathematical expression for displacement.
Given that displacement varies as square of time: \[ s \propto t^2 \Rightarrow s = kt^2 \] where \(k\) is a constant of proportionality.

Step 2: Find velocity by differentiating displacement. \[ v = \frac{ds}{dt} = \frac{d}{dt}(kt^2) = 2kt \] This shows velocity is directly proportional to time, hence velocity is not constant.

Step 3: Find acceleration by differentiating velocity. \[ a = \frac{dv}{dt} = \frac{d}{dt}(2kt) = 2k \] Since \(k\) is constant, acceleration is constant.

Step 4: Analyze options.
• Velocity is not constant → (A) incorrect
• Velocity changes with time → (B) partially true but not best answer
• Acceleration is constant → correct
• Acceleration does not change → (D) incorrect
• Momentum depends on velocity → not constant → (E) incorrect

Step 5: Final conclusion. \[ \boxed{\text{Acceleration is constant}} \]