A ball thrown up is caught by the thrower 6 sec after start. The height to which the ball has risen is : (g = 10 m/s^2)

Question

Accepted Answer

To determine the height to which the ball has risen, we need to analyze the motion of the ball using the principles of physics, specifically the equations of motion under uniform acceleration due to gravity.

1. **Understanding the Motion:**
   - The ball is thrown upwards and caught by the thrower after 6 seconds. This means the total time for the upward and downward journey is 6 seconds.
   - Since the time taken to go up is equal to the time taken to come down, the time taken to reach the maximum height is half of the total time, i.e., 3 seconds.

2. **Using the Equations of Motion:**
   - The formula to calculate the maximum height (H) reached by an object thrown upwards is given by:
     $$
     H = u \cdot t - \frac{1}{2} g \cdot t^2
     $$
     where $ u $ is the initial velocity, $ g $ is the acceleration due to gravity (10 m/s²), and $ t $ is the time taken to reach the maximum height.

3. **Calculating the Initial Velocity:**
   - At the maximum height, the final velocity is 0. Using the equation:
     $$
     v = u - g \cdot t
     $$
     Setting $ v = 0 $ and $ t = 3 $ seconds, we get:
     $$
     0 = u - 10 \cdot 3
     $$
     $$
     u = 30 \, \text{m/s}
     $$

4. **Calculating the Maximum Height:**
   - Substitute $ u = 30 \, \text{m/s} $, $ g = 10 \, \text{m/s}^2 $, and $ t = 3 \, \text{seconds} $ into the height equation:
     $$
     H = 30 \cdot 3 - \frac{1}{2} \cdot 10 \cdot 3^2
     $$
     $$
     H = 90 - \frac{1}{2} \cdot 10 \cdot 9
     $$
     $$
     H = 90 - 45
     $$
     $$
     H = 45 \, \text{m}
     $$

5. **Conclusion:**
   - The maximum height reached by the ball is 45 meters, which corresponds to Option 2.

6. **Explanation for Other Options:**
   - **Option 1 (30 m):** This height is incorrect because it does not account for the correct calculation of the initial velocity and time to reach the maximum height.
   - **Option 3 (60 m):** This height is incorrect as it overestimates the height based on the given time and gravitational acceleration.
   - **Option 4 (90 m):** This height is incorrect because it assumes the entire 6 seconds is spent reaching the maximum height, which is not the case.

Therefore, based on the calculations and logical reasoning, Option 2 (45 m) is indeed the correct answer.