A body falls freely from a height of 50 m. Simultaneously, another body is thrown from the surface of earth with a certain initial velocity. The two bodies meet at a height of 10 m. What is the initial velocity of the second body ?

Options

A. `4.9 m//s`
B. `9.8m//s`
C. `17.5m//s`
D. `19.6m//s `

Answer

To solve the problem, we need to analyze the motion of both bodies: the one falling freely from a height of 50 m and the one thrown upwards from the ground. We will use the equations of motion to find the initial velocity of the second body.

### Step-by-Step Solution:

1. **Identify the heights and distances:**
- The first body falls from a height of 50 m and meets the second body at a height of 10 m.
- Therefore, the first body falls a distance of \( 50 m - 10 m = 40 m \).

2. **Use the equation of motion for the first body:**
- The equation of motion for the first body (falling freely) is given by:
\[ s = ut + \frac{1}{2}gt^2 \]
- Here, \( s = 40 m \), \( u = 0 \) (initial velocity of the falling body), and \( g = 9.8 \, \text{m/s}^2 \).
- Substituting the values:
\[ 40 = 0 + \frac{1}{2} \cdot 9.8 \cdot t^2 \]
- Simplifying:
\[ 40 = 4.9t^2 \]
- Solving for \( t^2 \):
\[ t^2 = \frac{40}{4.9} \approx 8.1633 \]
- Taking the square root:
\[ t \approx \sqrt{8.1633} \approx 2.86 \, \text{s} \]

3. **Use the equation of motion for the second body:**
- The second body is thrown upwards with an initial velocity \( u \) and travels a distance of 10 m upwards.
- The equation of motion for the second body is:
\[ s = ut - \frac{1}{2}gt^2 \]
- Here, \( s = 10 m \), \( g = 9.8 \, \text{m/s}^2 \), and \( t \approx 2.86 \, \text{s} \).
- Substituting the values:
\[ 10 = u \cdot 2.86 - \frac{1}{2} \cdot 9.8 \cdot (2.86)^2 \]
- Calculating \( \frac{1}{2} \cdot 9.8 \cdot (2.86)^2 \):
\[ \frac{1}{2} \cdot 9.8 \cdot 8.1633 \approx 40 \]
- Therefore, we have:
\[ 10 = u \cdot 2.86 - 40 \]
- Rearranging gives:
\[ u \cdot 2.86 = 50 \]
- Solving for \( u \):
\[ u = \frac{50}{2.86} \approx 17.48 \, \text{m/s} \]

### Final Result:
The initial velocity of the second body is approximately \( 17.48 \, \text{m/s} \).