A particle of mass 2 / 3   kg with velocity v = - 15   m / s at t = - 2   s is acted upon by a force F = k - β t 2 . Here, k = 8   N and β = 2   N / s 2 . The motion is one-dimensional. Then, the speed at which the particle acceleration is zero again, is

Step-by-step solution

Force on the object is F = k - β t 2 Acceleration of the particle, a = F m = k - β t 2 m Acceleration is zero when k = β t 2 or t 2 = k β or t 2 = 8 2 ⇒ t = 2 s Now, a = d v d t = k - β t 2 m ⇒ d v = k - β t 2 m · d t Integrating between given limits, we have ⇒ ∫ v = - 15 v t = 2 s d v = ∫ t = - 2 s t = 2 s k - β t 2 m d t = 3 2 ∫ t = - 2 t = 2 8 - 2 t 2 d t v t = 2 s - - 15 = 3 2 8 t - 2 t 3 3 - 2 2 ⇒ v at t = 2 s = - 15 + 3 2 × 8 2 - - 2 - 2 3 8 - - 2 3 ⇒ v at t = 2 s = - 15 + 3 2 32 - 32 3 = - 15 + 3 2 64 3 = -