Romeo is chucking pebbles gently up to Juliet’s window, and he wants the pebbles to hit the window with only a horizontal compo nent of velocity. He is standing at the edge of a rose garden 8.0 m below her window and 9.0 m from the base of the wall (Fig. 3 55). How fast are the pebbles going when they hit her window?
The Correct Answer and Explanation is :
To solve this problem, we will break it down using the principles of projectile motion. We want to find the speed of the pebbles as they hit Juliet’s window, where they have only a horizontal component of velocity. The vertical component of the velocity at the point of impact is zero because the pebble is only moving horizontally when it reaches the window.
Given Information:
- The distance from Romeo to Juliet horizontally: (d_x = 9.0 \, \text{m})
- The vertical distance from Romeo to Juliet: (d_y = 8.0 \, \text{m})
- Acceleration due to gravity: (g = 9.8 \, \text{m/s}^2)
- The initial vertical velocity ((v_{iy})) is 0, as Romeo is throwing the pebbles horizontally.
Step 1: Vertical motion analysis
For the vertical motion, we use the kinematic equation for distance under constant acceleration:
[
d_y = v_{iy} t + \frac{1}{2} g t^2
]
Since (v_{iy} = 0) (the pebble is launched horizontally), the equation simplifies to:
[
d_y = \frac{1}{2} g t^2
]
Rearranging for (t), the time it takes for the pebble to fall 8 meters vertically:
[
t = \sqrt{\frac{2d_y}{g}} = \sqrt{\frac{2 \times 8.0}{9.8}} = \sqrt{1.6327} = 1.28 \, \text{seconds}
]
Step 2: Horizontal motion analysis
For the horizontal motion, we know that the horizontal velocity ((v_x)) remains constant (since there is no horizontal acceleration). We use the equation for horizontal displacement:
[
d_x = v_x t
]
We can solve for (v_x) as follows:
[
v_x = \frac{d_x}{t} = \frac{9.0}{1.28} = 7.03 \, \text{m/s}
]
Step 3: Final velocity
The pebble hits the window with only a horizontal velocity component. Thus, the speed at impact is simply the horizontal velocity:
[
v = v_x = 7.03 \, \text{m/s}
]
Conclusion:
The pebbles are traveling at a speed of 7.03 m/s when they hit Juliet’s window.
Explanation:
In this problem, Romeo is trying to launch the pebbles so they have only horizontal velocity when they hit the window. This means that, at the moment of impact, the vertical component of velocity is zero. To calculate the speed at which the pebble hits the window, we analyze both vertical and horizontal components of the pebble’s motion separately. First, we find how long it takes for the pebble to fall vertically from a height of 8.0 m. Using this time, we then calculate the horizontal velocity needed to cover the horizontal distance of 9.0 m in the same amount of time. The result is a speed of 7.03 m/s when the pebble strikes the window.