While standing at the edge of the roof of a building, you throw a stone upward with an initial speed of 7.53 m/s. The stone subsequently falls to the ground, which is 14.7 m below the point where the stone leaves your hand. At what speed does the stone impact the ground? How much time is the stone in the air? Ignore air resistance and take g = -9.80 m/s2.
h = ho + (V^2-Vo^2)/2g
h = 14.7 + (0-(7.53)^2)/-19.6 = 17.6 m.
Above gnd.
V^2 = Vo^2 + 2g*h
V^2 = 0 + 19.6*17.6 = 345
V = 18.6 m/s.
V = Vo + g*t
Tr = (V-Vo)/g = (0-7.53)/-9.8=0.768 s.=
Rise time.
h = Vo*t + 0.5g*t^2 = 17.6 m.
0 + 4.9t^2 = 17.6
t^2 = 3.59
Tf = 1.89 s. = Fall time.
T = Tr + Tf = 0.768 + 1.89 = 2.66 s. =
Time in air.
To find the speed at which the stone impacts the ground, we'll use the equations of motion.
Step 1: Determine the initial vertical velocity of the stone.
Given that the stone is thrown upward, the initial vertical velocity (v₀) is positive. Therefore, v₀ = 7.53 m/s.
Step 2: Calculate the time taken for the stone to reach its maximum height.
Using the equation of motion:
v = v₀ + gt, where v is the final vertical velocity, g is the acceleration due to gravity, and t is the time taken.
At the maximum height, the final vertical velocity will be 0 m/s (v = 0 m/s).
Therefore, 0 = 7.53 m/s + (-9.80 m/s²) * t.
Solving this equation for t:
9.80 m/s² * t = 7.53 m/s
t = 7.53 m/s / 9.80 m/s²
t ≈ 0.768 seconds
Step 3: Calculate the total time the stone is in the air.
Since the stone goes up and then falls back down, the total time in the air will be twice the time taken to reach the maximum height.
Therefore, total time = 2 * 0.768 seconds
total time ≈ 1.54 seconds
Step 4: Calculate the final vertical velocity when the stone hits the ground.
Using the equation of motion:
v = v₀ + gt
Substituting the values:
v = 7.53 m/s + (-9.80 m/s²) * 1.54 seconds
v ≈ -14.95 m/s
Since the velocity is negative, it indicates that the stone is moving downward. Taking the magnitude of this velocity, the speed at which the stone impacts the ground is approximately 14.95 m/s.
To find the speed of the stone when it impacts the ground, as well as the time it remains in the air, we can use the equations of motion. In this case, we need to consider the vertical motion of the stone.
Step 1: Determine the time it takes for the stone to reach its highest point.
The initial vertical velocity of the stone is 7.53 m/s, and it is thrown vertically upward. The final velocity at the highest point will be zero. The acceleration due to gravity is -9.80 m/s^2 (negative because it acts in the opposite direction to the motion).
We can use the equation of motion to find the time it takes to reach the highest point:
Final velocity squared = Initial velocity squared + 2 * acceleration * displacement
0 = (7.53 m/s)^2 + 2 * (-9.80 m/s^2) * displacement
Solving for displacement:
displacement = (7.53 m/s)^2 / (2 * 9.80 m/s^2)
displacement ≈ 2.88 m
Now we can use the equation of motion to find the time:
Final velocity = Initial velocity + acceleration * time
0 = 7.53 m/s + (-9.80 m/s^2) * time
Solving for time:
time = -7.53 m/s / (-9.80 m/s^2)
time ≈ 0.77 s
Step 2: Determine the time it takes for the stone to fall from the highest point to the ground.
The initial vertical velocity when the stone starts falling from the highest point is zero, as it is momentarily at rest. The displacement is 14.7 m (the height of the building). The acceleration due to gravity is -9.80 m/s^2.
We can use the equation of motion to find the time it takes to fall:
Final velocity squared = Initial velocity squared + 2 * acceleration * displacement
V^2 = 0^2 + 2 * (-9.80 m/s^2) * (-14.7 m)
V ≈ 17.14 m/s
The magnitude of the final velocity is 17.14 m/s, but since it is directed downward, the speed will also be positive 17.14 m/s.
Step 3: Determine the total time of flight.
The total time of flight is the sum of the time it takes to reach the highest point (0.77 s) and the time it takes to fall back down (which is also 0.77 s).
Total time of flight = 0.77 s + 0.77 s = 1.54 s
Therefore, the stone impacts the ground with a speed of approximately 17.14 m/s and remains in the air for about 1.54 seconds.