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 6.80 m. The stones are thrown with the same speed of 9.05 m/s. Find the location (above the base of the cliff) of the point where the stones cross paths.

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To find the location where the stones cross paths, we need to determine the time it takes for each stone to reach that point. Let's break down the problem step by step:

Step 1: Find the time it takes for the stone thrown upward to reach its highest point.
When the stone reaches its highest point, its vertical velocity becomes zero. We can use the kinematic equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.

At the highest point, the final velocity v is 0, the initial velocity u is 9.05 m/s (upward), and the acceleration a is -9.8 m/s^2 (due to gravity acting downward). Since the stone is thrown upward, the displacement is the height of the cliff, which is 6.8 m.

Using the equation, we can solve for the time t_1 it takes for the stone to reach its highest point:

0 = (9.05)^2 + 2(-9.8)t_1
0 = 82.1 - 19.6t_1
19.6t_1 = 82.1
t_1 = 82.1 / 19.6
t_1 ≈ 4.18 s

Step 2: Find the time it takes for the stone thrown downward to reach the location where the stones cross paths.
Since the stones are thrown simultaneously, the time it takes for both stones to cross paths will be the same. Therefore, we can use t_1 obtained in Step 1.

Step 3: Calculate the displacement of the stone thrown downward at the time it crosses paths with the other stone.
To find the displacement of the stone thrown downward, we can use the equation s = ut + (1/2)at^2, where u is the initial velocity, a is the acceleration, and t is the time.

The initial velocity of the stone thrown downward is -9.05 m/s, the acceleration due to gravity a is -9.8 m/s^2, and the time t_1 is 4.18 s.

Using the equation, we can solve for the displacement s_d of the stone thrown downward:

s_d = (-9.05)(4.18) + (1/2)(-9.8)(4.18)^2
s_d ≈ -37.78 m

Step 4: Calculate the location (above the base of the cliff) where the stones cross paths.
The location where the stones cross paths is the height of the cliff (6.8 m) plus the displacement of the stone thrown downward (s_d).

Location = height of cliff + displacement of stone thrown downward
Location = 6.8 + (-37.78)
Location ≈ -31.98 m

Therefore, the location where the stones cross paths is approximately 31.98 m below the base of the cliff.

To find the location where the two stones cross paths, we need to determine the time it takes for each stone to reach that point. We can then use the time to calculate the distance from the base of the cliff to the crossing point.

Let's start by considering the stone thrown upward. We can use the equation of motion for vertical motion:

h = ut + (1/2)gt^2

where h is the height, u is the initial velocity, g is the acceleration due to gravity (-9.8 m/s^2), and t is the time.

In this case, h is the height of the cliff (6.80 m), u is the initial velocity (9.05 m/s), and we need to solve for t.

6.80 = (9.05)t + (1/2)(-9.8)t^2

Simplifying this equation, we get a quadratic equation in terms of t:

-4.9t^2 + 9.05t + 6.80 = 0

Solving this equation will give us the time it takes for the stone thrown upward to reach the top of the cliff.

Now, let's consider the stone thrown downward. Since the initial velocity is in the opposite direction, it will have a negative value. Therefore, the equation of motion becomes:

h = ut + (1/2)gt^2

In this case, h is the height of the cliff (6.80 m), u is the initial velocity (-9.05 m/s), and we need to solve for t.

6.80 = (-9.05)t + (1/2)(-9.8)t^2

Again, simplifying this equation, we get:

-4.9t^2 - 9.05t + 6.80 = 0

Solving this equation will give us the time it takes for the stone thrown downward to reach the bottom of the cliff.

Once you have the two times, you can compare them to find the point at which the stones cross paths. Multiply the time by the initial velocity to get the distance traveled by each stone. The distance traveled by the stone thrown upward is the height of the cliff minus the distance traveled by the stone thrown downward.

If you solve the quadratic equations and find the two times, you can find the distance from the base of the cliff to the crossing point using the formula:

Crossing location = height of the cliff - distance traveled by the stone thrown downward.

Remember to consider whether the times are positive and whether they make sense in the context of the problem.