while standing at the edge of the roof of a building, you throw a stone upward with an initial speed of 7.79 m/s. the stone then falls to the ground 14.3 m below. At what speed does the stone impact the ground? how much time is it in the air? g=9.81 m/s^2 and ignore air resistance
V^2/2 increases by g*H m2/s^2, the potential energy loss.
The time in the air is the sum of the times spent going up and down.
The time going up is t1 = 7.79 m/s/g
The time spent coming down is given by
Vyo*t2 -(g/2)*t2^2 = -14.3 meters
Solve for t2
To find the speed at which the stone impacts the ground, we can use the concept of projectile motion.
First, let's analyze the vertical motion of the stone. Since the stone is thrown upward, it will experience free fall due to gravity while moving upward, and then undergo free fall again when falling back down.
The total distance covered by the stone, from the initial height to the ground, is the sum of the distance traveled upward and the distance traveled downward.
1. Calculate the time taken to reach the peak height:
We know that the vertical velocity at the highest point of the motion is zero (as the stone momentarily stops moving upward at the peak). Using this information, we can find the time it takes to reach the peak height. We can use the following formula:
v = u + at
Here, v = 0 m/s (final velocity), u = 7.79 m/s (initial velocity), a = -9.81 m/s^2 (acceleration due to gravity), and we need to solve for t (time taken).
Plugging in the values, we have:
0 = 7.79 - 9.81t
Rearranging the equation, we get:
9.81t = 7.79
t = 7.79 / 9.81
Calculating this, you can find that it takes approximately 0.79 seconds to reach the peak height.
2. Calculate the time taken to fall back to the ground:
The time taken to fall from the peak height to the ground is the same as the time taken to reach the peak height. So, the total time in the air is 2 times the time calculated in step 1.
Therefore, the total time in the air is 2 * 0.79 seconds, which equals 1.58 seconds.
3. Calculate the speed at impact:
To calculate the speed at impact, we need to determine the final vertical velocity. We can use the formula:
v = u + at
Here, v = ? (final velocity at impact), u = 0 m/s (initial velocity when the stone falls from the peak height), a = 9.81 m/s^2 (acceleration due to gravity), and t = 1.58 seconds (total time in the air).
Plugging in the values, we have:
v = 0 + (9.81 * 1.58)
Calculating this, you can find that the speed at which the stone impacts the ground is approximately 15.45 m/s.