Can you help with velocity of a boat please.?

After switching off the engine of a speedy motorboat, the acceleration of the boat follows the equation, color(red)(a=-Kv^3"; "K="constant". Show that , after t time from switching the engine off, the velocity of the boat is color(green)(v=v_0/(sqrt(1+2K cdot t cdot v_0 ^2)), v_0= initial velocity while switching the motor to the off position.

1 Answer
Apr 1, 2018

See below.

Explanation:

If a = dot v = -K v^3 then after multiplication by v in both sides we have

1/2d/(dt) v^2 = -K v^4 and making u = v^2 we get

1/2 d/(dt)u = -K u^2 This is a separable differential equation with solution

(du)/u^2 + 2K dt = 0 rArr u = 1/(2K t +C_0) and finally

u = v^2 rArr v = 1/sqrt(2K t + C_0)

now considering

v_0 = 1/sqrtC_0 we have

v = v_0/sqrt(1+2K v_0^2 t)