Question

A particle is moving along x-axis with its position ($ x $) varying with time ($ t $) as:  
$ x = \alpha t^4 + \beta t^2 + \gamma t + \delta. $  
The ratio of its initial velocity to its initial acceleration, respectively, is:

• A. \(2\alpha : \delta\)
• B. \(\gamma : 2\delta\)
• C. \(4\alpha : \beta\)
• D. \(\gamma : 2\beta\)

Answer 1

The Correct Option is D

Velocity (v) = $\frac{dx}{dt} = 4\alpha t^3 + 2\beta t + \gamma$ Initial velocity (at t = 0) = γ

Acceleration (a) = $\frac{dv}{dt} = 12\alpha t^2 + 2\beta$ Initial acceleration (at t = 0) = 2β

Ratio of initial velocity to initial acceleration = $\frac{\gamma}{2\beta}$