A ball experienced free fall and dropped into a lake full of water.The ball sunk then float back to the surface.What is the maximum depth that the ball can sink

Question

Density of wooden ball = 0.6×10³ kgm^-3

Free fall height = 5m above water surface

Density of water = 1000 kgm^-3

Gravitational acceleration = 10ms^-2

Velocity of ball when it hits the water surface = 10ms^-1

** water resistance is not considered* The answer is 7.5m btw

Answer 1

I would first find its retarding acceleration in the water, and then use a kinematic relation to find the distance where the speed is instantaneously zero, which in this problem is the maximum depth.

 

f = buoyant force

D_w = density of water

m = mass of ball

V = volume of ball = volume of displaced water

D_b = density of ball = m/V

h = maximum depth = ?

 

f = D_wVg

 

Assuming down direction is positive,

 

a = (mg - f)/m = g - f/m = g - D_wVg/m = g - (D_w/D_b)g = g(1 - D_w/D_b)

 

Kinematic relation:

 

v - v₀² = 2ah

 

v = instantaneous speed at max depth (at h) = 0 ⇒

 

h = -v₀²/(2a) = -v₀²/[2g(1 - D_w/D_b)]

 

v₀ = 10 m/s

 

h = -(10)² / [2(10)(1 - 1000/600)] m = 7.5 m