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 6.82 m. The stones are thrown with the same speed of 8.88 m/s. Find the location (above the base of the cliff) of the point where the stones cross paths.

I can't figure out how to manipulate free fall...please help!!!

the height of the falling stone is

6.82-8.8t-4.9t^2

the height of the stone thrown upward is

8.8t - 4.9t^2

So, when do they meet? At t=0.3875

Now plug that into either function to get the height at that time.

http://www.wolframalpha.com/input/?i=solve+6.82-8.8t-4.9t^2++%3D++8.8t+-+4.9t^2

To find the location where the stones cross paths, we need to determine the time it takes for each stone to reach that location.

Let's start by considering the stone thrown upward from the base of the cliff. Since it is thrown straight upward, its initial velocity is positive (+8.88 m/s). The acceleration due to gravity is always directed downward and has a magnitude of 9.8 m/s^2. At the highest point, the stone will have a velocity of 0 m/s and then start to fall downward.

To determine the time it takes for the stone to reach its highest point, we can use the equation:

Final velocity = Initial velocity + (acceleration * time)

0 m/s = 8.88 m/s - (9.8 m/s^2 * t1)

Solving for t1, we get:

-8.88 m/s = -9.8 m/s^2 * t1

t1 = -8.88 m/s / -9.8 m/s^2 = 0.905 s

So, it takes approximately 0.905 seconds for the stone to reach its highest point.

Next, to determine the total time of flight for the stone thrown upward, we can multiply the time to reach the highest point by 2:

Total time of flight (upward) = 0.905 s * 2 = 1.81 s

Now let's consider the stone thrown downward from the top of the cliff. Since it is thrown straight downward, its initial velocity is negative (-8.88 m/s). The acceleration due to gravity is still -9.8 m/s^2 in this case.

To determine the time it takes for the stone to fall to the location where it crosses paths with the other stone, we can use the equation:

Final position = Initial position + (initial velocity * time) + (0.5 * acceleration * time^2)

The initial position of the stone thrown downward is 6.82 m (the height of the cliff).

0 m = 6.82 m + (-8.88 m/s * t2) + (0.5 * -9.8 m/s^2 * t2^2)

Simplifying this equation and rearranging, we get:

4.9 t2^2 + 8.88 t2 - 6.82 = 0

Solving this quadratic equation, we find t2 to be approximately 0.671 seconds.

So, it takes approximately 0.671 seconds for the stone thrown downward to reach the location where the stones cross paths.

Now, to determine the location (above the base of the cliff) where the stones cross paths, we can use either stone's position at that time.

Let's use the stone thrown upward. Its position at time t = 0.671 seconds can be calculated using the equation:

Position = Initial position + (initial velocity * time) + (0.5 * acceleration * time^2)

Position = 0 m + (8.88 m/s * 0.671 s) + (0.5 * -9.8 m/s^2 * (0.671 s)^2)

Calculating this, we find the position to be approximately 3.79 meters above the base of the cliff.

Therefore, the stones cross paths at a height of approximately 3.79 meters above the base of the cliff.