An object of mass m is released from rest from the top of a smooth inclined plane of height h. Its speed at the bottom of the plane is proportional to

Options

A. `m^0`
B. `m`
C. `m^2`
D. `m^(-1)`

Explanation

To solve the problem, we will use the principle of conservation of energy. Here's a step-by-step breakdown of the solution:

### Step 1: Identify the Initial and Final States
- The object of mass \( m \) is released from rest at the top of the inclined plane (point A) with height \( h \).
- At the bottom of the inclined plane (point B), the object will have a certain speed \( V \).

### Step 2: Write the Energy Conservation Equation
According to the conservation of mechanical energy:
\[ \text{Potential Energy at A} + \text{Kinetic Energy at A} = \text{Potential Energy at B} + \text{Kinetic Energy at B} \]

### Step 3: Calculate the Energies
- At point A (top of the incline):
- Potential Energy (PE) = \( mgh \)
- Kinetic Energy (KE) = \( 0 \) (since the object is at rest)

- At point B (bottom of the incline):
- Potential Energy (PE) = \( 0 \) (height is zero)
- Kinetic Energy (KE) = \( \frac{1}{2} m V^2 \)

### Step 4: Set Up the Equation
Substituting the values into the energy conservation equation:
\[ mgh + 0 = 0 + \frac{1}{2} m V^2 \]
This simplifies to:
\[ mgh = \frac{1}{2} m V^2 \]

### Step 5: Simplify the Equation
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ gh = \frac{1}{2} V^2 \]

### Step 6: Solve for \( V \)
Rearranging the equation to solve for \( V \):
\[ V^2 = 2gh \]
Taking the square root of both sides gives:
\[ V = \sqrt{2gh} \]

### Step 7: Analyze the Proportionality
- The expression \( V = \sqrt{2gh} \) shows that the speed \( V \) is dependent on \( g \) (acceleration due to gravity) and \( h \) (height).
- Importantly, there is no dependence on the mass \( m \). Thus, the speed \( V \) is independent of the mass of the object.

### Conclusion
Since the speed \( V \) does not depend on \( m \), we can conclude that:
\[ V \propto m^0 \]
Thus, the answer is \( m^0 \).

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