Two stones are thrown from the top of a tower, one straight down with an initial speed u and the second straight up with the same speed u. When the two stones hit the ground they will have speeds in the ratio.

Options

A. `2:3`
B. `2:1`
C. `1:2`
D. `1:1`

Answer

To solve the problem of the two stones thrown from the top of a tower, we will analyze the motion of each stone separately and determine their final speeds when they hit the ground.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Stone 1 is thrown straight down with an initial speed \( u \).
- Stone 2 is thrown straight up with the same initial speed \( u \).

2. **Determine the Motion of Stone 1**:
- For Stone 1, which is thrown downward, we can use the kinematic equation:
\[ v_1^2 = u^2 + 2gh \]
Here, \( v_1 \) is the final velocity of Stone 1 when it hits the ground, \( g \) is the acceleration due to gravity, and \( h \) is the height of the tower.

3. **Determine the Motion of Stone 2**:
- For Stone 2, which is thrown upward, it will first move upward until it reaches its maximum height and then fall back down. The maximum height \( h' \) it reaches can be calculated using:
\[ 0 = u^2 - 2gh' \implies h' = \frac{u^2}{2g} \]
- After reaching the maximum height, Stone 2 will fall a total distance of \( h + h' \) to reach the ground. The total distance fallen is:
\[ h + h' = h + \frac{u^2}{2g} \]
- Using the kinematic equation for Stone 2 when it hits the ground:
\[ v_2^2 = u^2 + 2g\left(h + \frac{u^2}{2g}\right) \]
Simplifying this gives:
\[ v_2^2 = u^2 + 2gh + u^2 = 2u^2 + 2gh \]

4. **Calculate the Final Velocities**:
- From the equations derived:
- For Stone 1:
\[ v_1^2 = u^2 + 2gh \]
- For Stone 2:
\[ v_2^2 = 2u^2 + 2gh \]

5. **Find the Ratio of Final Velocities**:
- Now, we can find the ratio of the final speeds of the two stones:
\[ \frac{v_1}{v_2} = \frac{\sqrt{u^2 + 2gh}}{\sqrt{2u^2 + 2gh}} = \sqrt{\frac{u^2 + 2gh}{2u^2 + 2gh}} \]
- This simplifies to:
\[ \frac{v_1}{v_2} = \sqrt{\frac{1 + \frac{2gh}{u^2}}{2 + \frac{2gh}{u^2}}} \]

6. **Conclusion**:
- The ratio of the speeds when the stones hit the ground is given by:
\[ \frac{v_1}{v_2} = \sqrt{\frac{1 + \frac{2gh}{u^2}}{2 + \frac{2gh}{u^2}}} \]