A girl throws a ball at a vertical wall 4 m away (Figure). The ball is 2 m above the ground when it leaves the girl’s hand with an initial velocity of When the ball hits the wall, the horizontal component of its velocity is reversed; the vertical component remains unchanged. Where does the ball hit theground?
The Correct Answer and Explanation is :
Problem Breakdown:
- Initial conditions: A ball is thrown with an initial velocity at a height of 2 meters above the ground. The ball is thrown horizontally toward a vertical wall 4 meters away.
- When the ball hits the wall: The horizontal component of the ball’s velocity is reversed, but the vertical component remains unchanged.
- Goal: To find where the ball hits the ground after hitting the wall.
Steps for Solution:
- Horizontal Motion:
- The ball travels a horizontal distance of 4 meters to hit the wall.
- The horizontal velocity is constant throughout the flight because there are no forces acting horizontally (ignoring air resistance).
- Vertical Motion:
- The vertical motion is influenced by gravity, causing the ball to accelerate downward at (9.8 \, \text{m/s}^2).
- The vertical component of the ball’s velocity is unchanged when it hits the wall.
- Reversing Horizontal Velocity:
- Upon hitting the wall, the horizontal velocity reverses but retains the same magnitude.
- Time to Reach the Wall:
- The time (t_{\text{wall}}) to hit the wall is determined by the horizontal velocity ((v_{\text{horizontal}})) and the distance to the wall (4 meters):
[
t_{\text{wall}} = \frac{4}{v_{\text{horizontal}}}
]
- Vertical Displacement After Hitting the Wall:
- After hitting the wall, the ball will continue to fall vertically while moving horizontally in the opposite direction. The vertical displacement is given by the formula:
[
y_{\text{fall}} = \frac{1}{2} g t_{\text{fall}}^2
]
where (g = 9.8 \, \text{m/s}^2) is the acceleration due to gravity, and (t_{\text{fall}}) is the time it takes for the ball to reach the ground after hitting the wall.
- Where Will the Ball Hit the Ground?
- After the ball hits the wall, it moves horizontally in the opposite direction with the same velocity. Using the time it takes for the ball to fall, we can calculate how far the ball travels horizontally before hitting the ground. This distance will be the total horizontal distance from the point it was thrown to the point where it hits the ground.