Question

A body of mass \( m \) moves along the X-axis such that at time \( t \), its position is \( x(t) = \alpha t^4 - \beta t^3 + \gamma t \), where \( \alpha \), \( \beta \), and \( \gamma \) are constants. The acceleration of the body is:

• A. \( 24\alpha t^3 - 6\beta t \)
• B. \( \alpha t^2 - 6\beta t \)
• C. \( 6\alpha t^2 - 6\beta t \)
• D. \( 6\alpha t^3 - 6\beta t \)

Hint:

To find the acceleration from the position function, differentiate twice with respect to time.

Answer 1

The Correct Option is A

To find the acceleration, we first differentiate the position function twice with respect to time: \[ v(t) = \frac{d}{dt} \left( \alpha t^4 - \beta t^3 + \gamma t \right) = 4\alpha t^3 - 3\beta t^2 + \gamma \] \[ a(t) = \frac{d}{dt} \left( 4\alpha t^3 - 3\beta t^2 + \gamma \right) = 12\alpha t^2 - 6\beta t \] Thus, the acceleration is \( a(t) = 24\alpha t^3 - 6\beta t \).