A ball of relative density 0.8 falls into water from a height of 2 m. find the depth to which the ball will sink (neglect viscous forces)

Options

A. 8 m
B. 2 m
C. 6 m
D. 4 m

Answer

To solve the problem of determining how deep a ball of relative density 0.8 will sink in water after falling from a height of 2 meters, we can follow these steps:

### Step 1: Understand the Forces Acting on the Ball
When the ball is submerged in water, two main forces act on it:
- The weight of the ball (downward force)
- The buoyant force (upward force)

### Step 2: Calculate the Weight of the Ball
The weight of the ball (W) can be expressed as:
\[ W = mg \]
where \( m \) is the mass of the ball and \( g \) is the acceleration due to gravity.

### Step 3: Calculate the Buoyant Force
The buoyant force (Fb) acting on the ball when it is submerged is given by Archimedes' principle:
\[ F_b = \text{density of liquid} \times g \times \text{volume displaced} \]
Given that the relative density of the ball is 0.8, we can express the density of the ball as:
\[ \text{Density of ball} = 0.8 \times \text{density of water} \]

### Step 4: Set Up the Equation of Motion
Using Newton's second law, we can express the net force acting on the ball when it is submerged:
\[ mg - F_b = ma \]
Substituting the expressions for weight and buoyant force, we have:
\[ mg - \text{density of water} \times g \times V = ma \]

### Step 5: Relate Volume to Mass
The volume of the ball can be expressed in terms of its mass and density:
\[ V = \frac{m}{\text{density of ball}} \]
Substituting this into the buoyant force equation gives:
\[ F_b = \text{density of water} \times g \times \frac{m}{\text{density of ball}} \]

### Step 6: Substitute and Simplify
Substituting \( F_b \) into the equation of motion:
\[ mg - \text{density of water} \times g \times \frac{m}{\text{density of ball}} = ma \]
This simplifies to:
\[ mg \left(1 - \frac{\text{density of water}}{\text{density of ball}}\right) = ma \]

### Step 7: Calculate the Acceleration
From the relative density of the ball (0.8), we can find:
\[ a = g \left(1 - \frac{1}{0.8}\right) = g \left(1 - 1.25\right) = -\frac{g}{4} \]
The negative sign indicates that the acceleration is upward.

### Step 8: Calculate the Velocity Upon Impact
The velocity \( v \) of the ball just before it hits the water can be calculated using the equation:
\[ v = \sqrt{2gh} \]
Substituting \( h = 2 \) m:
\[ v = \sqrt{2g \cdot 2} = 2\sqrt{g} \]

### Step 9: Use the Kinematic Equation to Find Depth
Using the kinematic equation:
\[ v^2 = u^2 + 2as \]
where:
- \( v = 0 \) (final velocity at the maximum depth)
- \( u = 2\sqrt{g} \) (initial velocity)
- \( a = -\frac{g}{4} \) (acceleration)

Substituting these values:
\[ 0 = (2\sqrt{g})^2 + 2\left(-\frac{g}{4}\right)s \]
This simplifies to:
\[ 0 = 4g - \frac{g}{2}s \]
Rearranging gives:
\[ \frac{g}{2}s = 4g \]
Thus:
\[ s = 8 \text{ meters} \]

### Conclusion
The depth to which the ball will sink in water is **8 meters**.