Motion of a Shell Fired Vertically Upwards from a Moving Trolley

When a shell is fired vertically upwards from a trolley moving horizontally, the motion of the shell can be analyzed in two separate components: vertical motion and horizontal motion. Let's break down the problem and find the horizontal range of the shell.

Vertical Motion

The vertical motion of the shell can be analyzed independently since the horizontal and vertical components of motion are independent of each other. The shell is fired upwards with an initial velocity 'v' relative to the trolley.

- The initial vertical velocity of the shell is 'v' (upwards).
- The acceleration due to gravity is 'g' (downwards).
- The time taken by the shell to reach the maximum height can be calculated using the equation:
t = v/g
- At the maximum height, the vertical velocity becomes zero.
- The time taken by the shell to fall back to the ground can be calculated using the equation:
T = 2t = 2v/g
- The total time of flight can be calculated as the sum of the time taken to reach the maximum height and the time taken to fall back:
T = t + T = v/g + 2v/g = 3v/g

Horizontal Motion

The horizontal motion of the shell is influenced by the horizontal velocity of the trolley, denoted as 'v'.

- The horizontal velocity of the shell remains constant throughout its motion and is equal to the velocity of the trolley, 'v'.
- The horizontal distance covered by the shell can be calculated using the equation:
R = v' * T
- Substituting the value of T, we get:
R = v' * (3v/g)
R = (3v'v)/g

Therefore, the horizontal range of the shell is given by (3v'v)/g, which is the final answer.

Conclusion

When a shell is fired vertically upwards from a trolley moving horizontally, the horizontal range of the shell can be found by considering the independent vertical and horizontal components of motion. By analyzing the vertical motion and the horizontal motion separately, we can determine that the horizontal range of the shell is equal to (3v'v)/g.