A particle is moving along x-axis with its position (x) varying with time (t) as x = αt^4 + βt^2 + γt + δ. ​

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:

(1) \(2 \alpha: \delta\)

(2) \(\gamma: 2 \delta\)

(3) \(4 \alpha: \beta\)

(4) \( \gamma: 2 \beta\)

Answer 1

Correct option is : (4) \(\gamma: 2 \beta\)

Position of particle, \(x=\alpha t^{4}+\beta t^{2}+\gamma t+\delta\)

\( \text { Velocity } v=\frac{d x}{d t}=4 \alpha t^{3}+2 \beta t+\gamma \)

\(\text { Initial velocity }=v(t=0)=\gamma\) 

\( \text { Acceleration } a=\frac{d v}{d t}=12 \alpha t^{2}+2 \beta\) 

\(\text { Initial acceleration }=a(t=0)=2 \beta\) 

\(\therefore \frac{v(t=0)}{a(t=0)}=\frac{\gamma}{2 \beta}\)