At t = 0 s time a stone is thrown from the top of a cliff above a
lake. Another stone is thrown down from the same point , 1.6 s later , with a
initial velocity of 32.0 m / s. The two stones hit the water in one instant.
Determine the cliff height.
To determine the cliff height, we can use the equation of motion for vertically thrown objects. Here are the steps to find the solution:
Step 1: Identify the known values:
- The time delay between the two stones: Δt = 1.6 s
- The initial velocity of the second stone: v2 = 32.0 m/s
- The acceleration due to gravity: g = 9.8 m/s^2
Step 2: Calculate the time it takes for the second stone to hit the water:
Since the second stone is thrown 1.6 seconds after the first stone, its time of flight is also 1.6 seconds less than the first stone's time of flight. Therefore, the time it takes for the second stone to hit the water is t2 = t1 - Δt.
Step 3: Calculate the time it takes for the first stone to hit the water:
Since we know both stones hit the water at the same time, we can set t1 = t2.
Step 4: Use the equation of motion to determine the height of the cliff:
The equation for the height h at time t (when the stone hits the water) is given by: h = v0t - (1/2)gt^2, where v0 is the initial velocity.
Step 5: Substitute the known values into the equation of motion:
For the first stone, h1 = v1*t1 - (1/2)g*t1^2.
For the second stone, h2 = v2*t2 - (1/2)g*t2^2.
Step 6: Set h1 = h2 and solve for the cliff height h:
v1*t1 - (1/2)g*t1^2 = v2*t2 - (1/2)g*t2^2.
Substituting t1 = t2, we get:
v1*t2 - (1/2)g*t2^2 = v2*t2 - (1/2)g*t2^2.
After canceling out the common terms, we have:
v1*t2 = v2*t2.
Solving for t2, we get:
t2 = (v1 * t2) / v2.
Step 7: Solve for the cliff height h:
Using our value of t2 from the equation above, substitute it into the equation for the first stone's height:
h = v1*t1 - (1/2)g*t1^2.
Step 8: Substitute the value of t1 = t2 into the equation for the cliff height h:
h = v1*(t2) - (1/2)g*(t2)^2.
Step 9: Substitute the known values:
- v1 = 0 (since the first stone is thrown vertically upward)
- v2 = 32.0 m/s
- g = 9.8 m/s^2.
Step 10: Solve for the cliff height h.
By following these steps, you can find the height of the cliff.
To determine the cliff height, we need to find the time it took for the first stone to hit the water and use that time to calculate the height.
Let's first find the time it took for the first stone to hit the water.
We know that the second stone was thrown 1.6 seconds after the first stone, so we can assume that the first stone took 1.6 seconds to hit the water.
Now, let's find the height using the kinematic equation for vertical motion:
h = v0 * t + (1/2) * g * t^2
Where:
h is the height (cliff height in this case)
v0 is the initial velocity of the stone (which we don't know)
t is the time taken for the stone to hit the water (1.6 seconds)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since the stones hit the water at the same time, the time for both stones to hit the water is the same. By using this information, we can set up an equation to find the height of the cliff:
h = v0 * t + (1/2) * g * t^2
Substituting the known values:
0 = v0 * 1.6 + (1/2) * 9.8 * (1.6)^2
We can solve this equation to find the initial velocity of the stone:
0 = 1.6v0 + 1/2*9.8*2.56
-1.6v0 = 1.568
v0 = -0.98 m/s
The negative sign indicates that the initial velocity is in the downward direction.
Now that we have the initial velocity of the stone (-0.98 m/s), we can find the height using the same kinematic equation:
h = v0 * t + (1/2) * g * t^2
h = -0.98 * 1.6 + (1/2) * 9.8 * (1.6)^2
Evaluating the equation, we find:
h = -1.568 + 12.544
h ≈ 10.976
Therefore, the height of the cliff is approximately 10.976 meters.