The acceleration of a particle depends on its speed according to a = k(vₜ²-v²)/v, where k>0 and vₜ is a constant. At x=0 its speed is vₜ/2. The displacement required for its speed to become 3vₜ/4 is:
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Question

The acceleration of a particle depends on its speed according to a = k(vₜ²-v²)/v, where k>0 and vₜ is a constant. At x=0 its speed is vₜ/2. The displacement required for its speed to become 3vₜ/4 is:

Options

G1 Physics · Accepted Answer

Use a = v dv/dx. Hence v dv/dx = k(vₜ²−v²)/v ⇒ dx/dv = v²/[k(vₜ²−v²)]. Let y=v/vₜ. Then dx = (vₜ/k) y²/(1−y²) dy. Now y²/(1−y²) = −1 + 1/(1−y²). Therefore x = (vₜ/k) ∫₁ᐟ₂→³ᐟ₄ [−1 + 1/(1−y²)]dy. Using ∫dy/(1−y²) = (1/2)ln[(1+y)/(1−y)], we get x = (vₜ/k) [−y + (1/2)ln((1+y)/(1−y))]₁ᐟ₂→³ᐟ₄ = (vₜ/k) [−1/4 + (1/2)ln(7/3)]. Therefore option B is correct.

The acceleration of a particle depends on its speed according to a = k(vₜ²−v²)/v, where k>0 and vₜ is a constant. At x=0 its speed is vₜ/2. The displacement required for its speed to become 3vₜ/4 is: