A ball is thrown upwards, it takes 4 s to reach back to the ground. Find its initial velocity

Options

A. 30 m/s
B. 10 m/s
C. 40 m/s
D. 20 m/s

Answer

To solve the problem of finding the initial velocity of a ball thrown upwards that takes 4 seconds to return to the ground, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**:
The total time of flight for the ball is given as \( t = 4 \) seconds. Since the ball is thrown upwards and then comes back down, the time taken to reach the maximum height is half of the total time. Therefore, the time to reach the maximum height is:
\[ t_{\text{up}} = \frac{t}{2} = \frac{4}{2} = 2 \text{ seconds} \]

2. **Use the Equation of Motion**:
We can use the equation of motion for vertical displacement:
\[ s = ut - \frac{1}{2}gt^2 \]
where:
- \( s \) = displacement (which is 0 for the entire journey since it returns to the starting point),
- \( u \) = initial velocity (what we want to find),
- \( g \) = acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( t \) = total time of flight (4 seconds).

3. **Set Up the Equation**:
Since the displacement \( s = 0 \) for the entire motion, we can set up the equation:
\[ 0 = u(4) - \frac{1}{2}g(4^2) \]
Substituting \( g = 10 \, \text{m/s}^2 \):
\[ 0 = 4u - \frac{1}{2}(10)(16) \]
Simplifying this gives:
\[ 0 = 4u - 80 \]

4. **Solve for Initial Velocity \( u \)**:
Rearranging the equation to solve for \( u \):
\[ 4u = 80 \]
\[ u = \frac{80}{4} = 20 \, \text{m/s} \]

5. **Conclusion**:
The initial velocity of the ball is \( 20 \, \text{m/s} \).