A stone is dropped from the roof of a high building. A second stone is dropped 1.50 s later. How far apart are the stones when the second one has reached a speed of 12.0 m/s?
The correct answer and explanation is :
The two stones are 29.04 meters apart when the second stone reaches a speed of 12.0 m/s.
Explanation:
- Understanding the motion:
- The stones are in free fall, meaning they only experience acceleration due to gravity (( g = 9.81 \, m/s^2 )).
- The first stone is dropped from rest.
- The second stone is dropped 1.5 seconds later.
- Finding time for the second stone:
- The second stone accelerates uniformly under gravity. Using the equation for velocity:
[
v = g t
]
Substituting ( v = 12.0 \, m/s ):
[
12.0 = 9.81 t_2
]
Solving for ( t_2 ):
[
t_2 = \frac{12.0}{9.81} = 1.22 \text{ s}
]
- Finding distance fallen:
- The displacement of a freely falling object is given by:
[
d = \frac{1}{2} g t^2
] - For the second stone:
[
d_2 = \frac{1}{2} (9.81) (1.22)^2
]
[
d_2 = 7.32 \text{ m}
] - The first stone has fallen for ( t_1 = 1.22 + 1.5 = 2.72 ) seconds:
[
d_1 = \frac{1}{2} (9.81) (2.72)^2
]
[
d_1 = 36.36 \text{ m}
]
- Finding the separation:
[
\text{Distance apart} = d_1 – d_2
]
[
= 36.36 – 7.32
]
[
= 29.04 \text{ m}
]
Thus, when the second stone reaches a speed of 12.0 m/s, the two stones are 29.04 meters apart.