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 6.86 m. The stones are thrown with the same speed of 9.78 m/s. Find the location (above the base of the cliff) of the point where the stones cross paths.
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. Let's call the time it takes for the stone thrown upward to reach the crossing point "t" and the time it takes for the stone thrown downward to reach the crossing point "T".
We can use the equations of motion to solve for these times. Since both stones are thrown with the same speed of 9.78 m/s, their initial velocities are also the same, but in opposite directions. The velocity of the stone thrown upward is positive 9.78 m/s, and the velocity of the stone thrown downward is negative 9.78 m/s.
The equation for the height of an object in free fall is given by h = v0t + (1/2)gt^2, where:
- h is the height of the object
- v0 is the initial velocity of the object
- t is the time
- g is the acceleration due to gravity, which is approximately -9.8 m/s^2 when taking downward as the negative direction
For the stone thrown upward:
h = 6.86 m (height of the cliff) and v0 = 9.78 m/s (upward velocity)
We can plug these values into the equation and solve for t.
6.86 = 9.78t - (1/2)(9.8)t^2
This is a quadratic equation that we can solve for t using any suitable method, such as factoring, completing the square, or using the quadratic formula.
After finding the value of t, we can substitute it into the equation for the stone thrown downward. Since we are looking for the height above the base of the cliff, we will subtract the height of the cliff from the downward stone's position. The equation is similar to the one before, but with a negative initial velocity and a positive height of (6.86 m).
-(6.86) = -9.78T + (1/2)(9.8)T^2
Again, we can solve this quadratic equation to find the value of T.
Once we have the values of t and T, we can determine the location where the stones cross paths. This is done by calculating the position of either stone at the corresponding time. The position is given by s = v0t + (1/2)gt^2 or s = v0T + (1/2)gT^2, depending on which stone's position we calculate.
Note: It's important to be careful with the signs of the quantities involved to ensure accuracy in the calculations.