A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

Step-by-step solution

Consider a small steel ball as shown below: As shown in diagram, buoyancy force F B = F . Applying Newton's second law, we get m g - F B - F v = m a ⇒ ρ 4 3 π r 3 g - ρ L 4 3 π r 3 g - 6 π η r v = m d v d t Let 4 3 m π R 3 g ρ - ρ L = K 1 and 6 π η r m = K 2 , then we get ⇒ d v d t = K 1 - K 2 v Integrating both sides, we get ⇒ ∫ 0 v d v K 1 - K 2 v = ∫ 0 t d t ⇒ - 1 K 2 ln K 1 - K 2 v 0 v = t ⇒ ln K 1 - K 2 v K 1 = - K 2 t ⇒ K 1 - K 2 v = K 1 e - K 2 t ⇒ v = K 1 K 2 1 - e - K 2 t Clearly, option B matches with the