A ball with mass m projected horizontally off the end of a table with an initial kinectic energy k. At a time t after it leaves the end of the table it has kinectic energy 3k.The time t is `n/gsqrt(k/m).Find the value of n Neglect air resistance.

Explanation

To solve the problem, we need to analyze the motion of the ball after it is projected horizontally off the table.

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- The ball is projected horizontally with an initial kinetic energy \( k \).
- The initial kinetic energy can be expressed as:
\[ k = \frac{1}{2} m v_x^2 \]
where \( m \) is the mass of the ball and \( v_x \) is the horizontal component of the velocity.

2. **Kinetic Energy After Time \( t \)**:
- At time \( t \), the kinetic energy of the ball is given as \( 3k \).
- The total kinetic energy at this time can be expressed as:
\[ 3k = \frac{1}{2} m (v_x^2 + v_y^2) \]
where \( v_y \) is the vertical component of the velocity after time \( t \).

3. **Relating the Two Kinetic Energies**:
- From the initial kinetic energy:
\[ k = \frac{1}{2} m v_x^2 \implies v_x^2 = \frac{2k}{m} \]
- Substituting this into the equation for \( 3k \):
\[ 3k = \frac{1}{2} m \left( \frac{2k}{m} + v_y^2 \right) \]
- Simplifying this gives:
\[ 3k = k + \frac{1}{2} m v_y^2 \]
- Rearranging yields:
\[ 2k = \frac{1}{2} m v_y^2 \implies v_y^2 = \frac{4k}{m} \]

4. **Finding the Vertical Velocity**:
- The vertical velocity \( v_y \) can be expressed as:
\[ v_y = \sqrt{\frac{4k}{m}} = 2\sqrt{\frac{k}{m}} \]

5. **Using Kinematics to Relate Time and Velocity**:
- The vertical motion of the ball is influenced by gravity. The vertical velocity \( v_y \) after time \( t \) can also be expressed using the equation of motion:
\[ v_y = g t \]
- Equating the two expressions for \( v_y \):
\[ g t = 2\sqrt{\frac{k}{m}} \]

6. **Solving for Time \( t \)**:
- Rearranging gives:
\[ t = \frac{2\sqrt{\frac{k}{m}}}{g} \]

7. **Expressing Time in Given Form**:
- The problem states that \( t = \frac{n}{g\sqrt{\frac{k}{m}}} \).
- Setting the two expressions for \( t \) equal to each other:
\[ \frac{2\sqrt{\frac{k}{m}}}{g} = \frac{n}{g\sqrt{\frac{k}{m}}} \]
- Cross-multiplying gives:
\[ 2\left(\sqrt{\frac{k}{m}}\right)^2 = n \]
- Since \( \left(\sqrt{\frac{k}{m}}\right)^2 = \frac{k}{m} \):
\[ n = 2 \cdot \frac{k}{m} \]

8. **Finding the Value of \( n \)**:
- From the previous steps, we find \( n = 2 \).

### Final Answer:
The value of \( n \) is \( 2 \).