For t≥0, the position of a particle is x = At²/(t²+τ²), where A and τ are positive constants. The maximum instantaneous velocity is:
Options

Question

For t≥0, the position of a particle is x = At²/(t²+τ²), where A and τ are positive constants. The maximum instantaneous velocity is:

G1 Physics · Accepted Answer

Differentiate: v = dx/dt = 2Aτ²t/(t²+τ²)². Let y=t/τ. Then v = (2A/τ) y/(1+y²)². Maximize f(y)=y/(1+y²)². df/dy = (1−3y²)/(1+y²)³. Thus y=1/√3 ⇒ t=τ/√3. At this value, f = (1/√3)/(4/3)² = 3√3/16. Therefore vₘₐₓ = (2A/τ)(3√3/16) = 3√3A/(8τ). Hence option C is correct.