Question - Buoyancy and Free Fall in Fluids Problem

1) Archimedes' principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The volume of the car submerged, \( V_c \), can be calculated from the given density of the car, \( \rho_c = 50 \) kg/m³, and the weight of the car, \( W_c = 980 \) N. Using the equation \( W_c = \rho_c g V_c \), where \( g \) is the acceleration due to gravity (\( 9.8 \) m/s²), we can find the volume of the submerged part of the car:

\[ V_c = \frac{W_c}{\rho_c g} = \frac{980}{50 \times 9.8} = 2 \text{ m}^3 \]

Since 10% of the car is above water, the total volume of the car, \( V_t \), is \( V_t = \frac{V_c}{0.9} \). Therefore,

\[ V_t = \frac{2}{0.9} \approx 2.22 \text{ m}^3 \]

The buoyant force \( F_b \) on the car is equal to the weight of the water displaced, which is \( \rho_w g V_c \), where \( \rho_w = 1000 \) kg/m³ is the density of water. Thus:

\[ F_b = \rho_w g V_c = 1000 \times 9.8 \times 2 = 19600 \text{ N} \]

2) For the free-falling body, the velocity \( v \) at which it hit the bottom of the pond can be found using the kinematic equation \( v^2 = u^2 + 2as \), where \( u \) is the initial velocity, \( a \) is the acceleration, and \( s \) is the distance. In water, the acceleration is less due to upward buoyancy and resistance, \( a' = g - (\rho_w/\rho_s)g - 0.4 \) where \( \rho_s \) is the density of substance. If the body's density is \( \rho_b = 550 \) kg/m³, then:

\[ a' = 9.81 - (\frac{1000}{550} \times 9.81) - 0.4 \]

\[ a' = 9.81 - 17.84 - 0.4 \]

\[ a' = -8.43 \text{ m/s}^2 \] (This is the effective acceleration considering buoyancy and resistance.)

Using the kinematic equation with \( u = 0 \) (starting from rest), \( a' \) as the effective acceleration, and \( s = 100 \) m (distance to the bottom of the pond):

\[ v^2 = 0 + 2(-8.43)(100) \]

\[ v^2 = -1686 \]

This yields an imaginary number which is not possible in real-life scenarios, which indicates that with the given parameters, the body would not reach the bottom due to the net upward acceleration. This suggests that an error might have been made in determining the resistance or the acceleration is not constant all the way down. In real-world scenarios, the object might eventually reach a terminal velocity where the net acceleration is zero.