A lead sphere is dropped into a medium. As the sphere falls, the velocity of lead sphere

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Answer

To solve the problem of a lead sphere falling into a medium, we need to analyze the forces acting on the sphere and how they affect its motion. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Forces Acting on the Sphere
When the lead sphere is dropped into the medium, three main forces act on it:
1. **Gravitational Force (Weight)**: This force acts downward and is given by \( F_g = mg \), where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity.
2. **Buoyant Force**: This force acts upward and is equal to the weight of the fluid displaced by the sphere. It is given by Archimedes' principle as \( F_b = \rho_{fluid} V_{sphere} g \), where \( \rho_{fluid} \) is the density of the fluid and \( V_{sphere} \) is the volume of the sphere.
3. **Drag Force**: As the sphere falls, it experiences a drag force that opposes its motion. This force depends on the velocity of the sphere and is typically modeled as \( F_d = -kv \), where \( k \) is a constant that depends on the properties of the fluid and \( v \) is the velocity of the sphere.

### Step 2: Write the Equation of Motion
The net force acting on the sphere can be expressed as:
\[ F_{net} = F_g - F_b - F_d \]
Using Newton's second law, we can write:
\[ ma = mg - \rho_{fluid} V_{sphere} g - kv \]
where \( a \) is the acceleration of the sphere.

### Step 3: Analyze the Motion Over Time
Initially, when the sphere is dropped, it will accelerate downwards due to the gravitational force being greater than the sum of the buoyant force and the drag force. As the sphere gains velocity, the drag force increases, which reduces the net force acting on the sphere.

### Step 4: Determine Terminal Velocity
Eventually, the sphere will reach a point where the net force becomes zero, meaning the acceleration will be zero. This occurs when:
\[ mg - \rho_{fluid} V_{sphere} g - kv_{terminal} = 0 \]
At this point, the sphere falls with a constant velocity known as the terminal velocity \( v_{terminal} \).

### Step 5: Conclusion
The velocity of the lead sphere will initially increase as it falls, but as the drag force increases, the acceleration will decrease until it reaches zero. At this point, the sphere will continue to fall at a constant terminal velocity.

### Final Answer
The velocity of the lead sphere first increases and then becomes constant as it reaches terminal velocity.

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