A small rock is thrown vertically upward with a speed of 17.0m/s from the edge of the roof of a 22.0m tall building. The rock doesn't hit the building on its way back down and lands in the street below. Air resistance can be neglected.
1. what is the speed of the rock just before it hits the street?
2. How long after the rock is thrown will it hit the street?
h(t) = 22.0 + 17.0t - 4.9t^2
v(t) = 17.0 - 9.8t
So, find t when h=0, and use that to find v at that instant.
To solve this problem, we can use the equations of motion for a vertically thrown object, assuming no air resistance.
1. To find the speed of the rock just before it hits the street:
First, we need to find the time it takes for the rock to reach the highest point of its trajectory. We can use the equation:
v = u + at
Where:
v = final velocity (0 m/s when the rock reaches the highest point)
u = initial velocity (17.0 m/s)
a = acceleration (due to gravity, which is -9.8 m/s^2)
t = time
Rearranging the equation to solve for time (t):
t = (v - u) / a
Substituting the values:
t = (0 - 17.0) / -9.8
t ≈ 1.73 s
Now, we know that the total time for the rock to travel up and down is double the time it took to reach the highest point. So the total time will be:
2t = 2 * 1.73 s
t_total ≈ 3.46 s
Once we have the total time of flight, we can find the final velocity of the rock just before it hits the street by using the equation:
v = u + at
Where:
v = final velocity (what we want to find)
u = initial velocity (17.0 m/s)
a = acceleration (due to gravity, which is -9.8 m/s^2)
t = time (total time of flight)
Substituting the values:
v = 17.0 + (-9.8) * 3.46
v ≈ -0.68 m/s
It's worth noting that the negative sign indicates that the final velocity is in the opposite direction of the initial velocity (downwards in this case). However, since we are looking for the speed (magnitude) and not the direction, we can ignore the negative sign. Therefore, the speed of the rock just before it hits the street is approximately 0.68 m/s.
2. To find how long after the rock is thrown it will hit the street:
We know that the total time of flight is approximately 3.46 seconds. Therefore, the time it takes for the rock to hit the street after it is thrown is also approximately 3.46 seconds.