Romeo is throwing pebbles gently up to Juliet’s window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 8.0 m below her window and 8.5 m from the base of the wall (Fig. 3–49). How fast are the pebbles going when they hit her window?
The Correct Answer and Explanation is :
To determine the velocity of the pebbles when they hit Juliet’s window with only a horizontal component of velocity, we can break this down into two main parts: the vertical and horizontal motion of the pebble.
Given:
- Vertical distance (height from Romeo to Juliet’s window) = 8.0 m.
- Horizontal distance = 8.5 m.
- Acceleration due to gravity, ( g = 9.8 \, m/s^2 ).
- Initial vertical velocity ( v_{y0} = 0 ) m/s (since Romeo is throwing the pebble horizontally).
Step 1: Vertical motion
Since the pebbles hit the window with no vertical velocity component, the pebble must first fall and then reverse its trajectory by only having a horizontal component upon impact. This implies the vertical velocity is zero at the time the pebble reaches the window.
For vertical motion, we can use the following kinematic equation:
[
y = v_{y0} t + \frac{1}{2} g t^2
]
Given that ( v_{y0} = 0 ) and ( y = 8.0 \, m ), we have:
[
8.0 = \frac{1}{2} g t^2
]
Solving for ( t ) (time to hit the window):
[
8.0 = \frac{1}{2} (9.8) t^2
]
[
t^2 = \frac{8.0 \times 2}{9.8}
]
[
t^2 = 1.6327 \quad \Rightarrow \quad t = 1.278 \, s
]
Step 2: Horizontal motion
Now, using the time ( t = 1.278 \, s ), we can calculate the horizontal velocity required for the pebble to travel a horizontal distance of 8.5 m. Since there is no horizontal acceleration (assuming no air resistance), we use the equation:
[
x = v_x t
]
where ( x = 8.5 \, m ), and solving for ( v_x ):
[
v_x = \frac{x}{t} = \frac{8.5}{1.278} = 6.65 \, m/s
]
Conclusion:
Thus, the horizontal velocity of the pebbles when they hit the window is 6.65 m/s. This means that Romeo must throw the pebbles with a speed of 6.65 m/s in the horizontal direction to ensure they reach Juliet’s window with only a horizontal velocity component.
This solution assumes that the pebble is only subject to gravity in the vertical direction and that air resistance is negligible.