Two stones are thrown simultaneously, one straight upward from the base of a cliff and the other straight downward from the top of the cliff. The height of the cliff is 5.28 m. The stones are thrown with the same speed of 9.33 m/s. Find the location (above the base of the cliff) of the point where the stones cross paths.
We will take velocity and distance as positive upwards, and distance = 0 at the bottom of the cliff.
Use
Distance = Vi*t-(1/2)gt²
For the stone thrown upwards,
X1(t)=9.33t-(1/2)gt^2
For the stone thrown downwards,
X2(t)=5.28-9.33t-(1/2)gt^2
When they meet, X1(t)=X2(t)
and solve for t.
For checking your answer, 0.25<t<0.3 s.
To find the location where the stones cross paths, we can determine the time it takes for each stone to reach that point and calculate the distance covered by each stone during that time.
Let's start by analyzing the motion of each stone individually.
Stone thrown upwards:
- Initial velocity (u) = 9.33 m/s (upward)
- Initial position (s) = 0 m (base of the cliff)
- Height at which they cross paths (h) = ?
Stone thrown downwards:
- Initial velocity (u) = 9.33 m/s (downward)
- Initial position (s) = 5.28 m (top of the cliff)
- Height at which they cross paths (h) = ?
Now, we can use the equations of motion to calculate the time it takes for each stone to reach the crossing point.
For the stone thrown upwards:
The equation for calculating displacement (s) in terms of initial velocity (u), time (t), and acceleration (a) is:
s = ut + (1/2)at^2
Since the stone is thrown upwards, the acceleration due to gravity (a) will be negative (-9.8 m/s^2) because it opposes the upward velocity. We know the initial position (s) is 0 m, so the equation becomes:
0 = (9.33 m/s)t - (1/2)(9.8 m/s^2)t^2
Simplifying and rearranging the equation, we get:
4.9t^2 - 9.33t = 0
This equation can be factored as follows:
t(4.9t - 9.33) = 0
So, either t = 0 (which is not relevant in our case) or 4.9t - 9.33 = 0.
Solving the equation, we find:
4.9t = 9.33
t = 9.33 / 4.9
t ≈ 1.9 s
Therefore, the stone thrown upwards reaches the crossing point in approximately 1.9 seconds.
Now, let's find the time it takes for the stone thrown downwards to reach the crossing point.
Using the same equation of motion, but with the negative value for the acceleration (-9.8 m/s^2) because the stone is thrown downwards, we have:
s = ut + (1/2)at^2
5.28 = (9.33 m/s)t + (1/2)(-9.8 m/s^2)t^2
Simplifying and rearranging the equation, we get:
-4.9t^2 + 9.33t + 5.28 = 0
Solving this quadratic equation, we find t ≈ 1.5 s and t ≈ 0.66 s. Since the stone cannot reach a time before it is actually thrown, we discard the negative value. Therefore, the stone thrown downwards reaches the crossing point in approximately 1.5 seconds.
Now that we know the time it takes for each stone to reach the crossing point, we can calculate the distances covered by each stone during that time.
For the stone thrown upwards:
Distance covered = Speed × Time
Distance covered = 9.33 m/s × 1.9 s
Distance covered ≈ 17.7 m
For the stone thrown downwards:
Distance covered = Speed × Time
Distance covered = 9.33 m/s × 1.5 s
Distance covered ≈ 14 m
Therefore, at the crossing point, the stone thrown upwards has covered a distance of approximately 17.7 meters above the base of the cliff, while the stone thrown downwards has covered a distance of approximately 14 meters above the base of the cliff.
Hence, the location (above the base of the cliff) where the stones cross paths is approximately 17.7 meters.