An arrow is shot straight up from the edge of a cliff with an initial velocity of 50 ft/s. Answer the following questions using English units. The acceleration due to gravity has a magnitude of 32.2 ft/s.
A. Find the time to reach the highest point. (please use step by step answers)
B. Find the displacement at the highest point.( please us step by step answers)
C. Find the height when the velocity is 25 ft/s.( please use step by step answers)
A. V = Vo + g*t. V = 0, Vo = 50/Ft/s, g = -32.2 F/s^2, t = ?.
B. h = Vo*t + 0.5g*t^2. t = Value calculated in part A.
C. V^2 = Vo^2 + 2g*h. V = 25 Ft/s, Vo = 50 Ft/s, h = ?.
A. To find the time taken to reach the highest point, we can use the kinematic equation:
vf = vi + at
In this case, the final velocity (vf) at the highest point is 0 ft/s (since the arrow will momentarily stop before falling back down), the initial velocity (vi) is 50 ft/s, and acceleration (a) is -32.2 ft/s² (negative because it acts opposite to the direction of the initial velocity).
Plugging in these values into the equation, we have:
0 = 50 + (-32.2)t
Now, let's solve for t:
-32.2t = -50
Divide both sides by -32.2:
t = -50 / -32.2
t ≈ 1.55 seconds
Therefore, the time taken to reach the highest point is approximately 1.55 seconds.
B. The displacement at the highest point can be found using the kinematic equation:
d = vit + (1/2)at²
Since the arrow comes to a stop at the highest point, its final velocity (vf) is 0 ft/s, initial velocity (vi) is 50 ft/s, and acceleration (a) is -32.2 ft/s². We already know the time (t) is approximately 1.55 seconds from the previous calculation.
Plugging in these values, we have:
d = (50)(1.55) + (1/2)(-32.2)(1.55)²
Simplifying:
d = 77.5 - 39.96325
d ≈ 37.54 feet
Therefore, the displacement at the highest point is approximately 37.54 feet.
C. To find the height when the velocity is 25 ft/s, we need to use the kinematic equation:
vf = vi + at
Again, the final velocity (vf) is 25 ft/s, initial velocity (vi) is 50 ft/s, and acceleration (a) is -32.2 ft/s². We need to find the time (t).
Plugging in these values, we have:
25 = 50 + (-32.2)t
-32.2t = 25 - 50
-32.2t = -25
Divide both sides by -32.2:
t = -25 / -32.2
t ≈ 0.776 seconds
Now that we know the time, we can find the height using the following kinematic equation:
d = vit + (1/2)at²
Plugging in the values, we have:
d = (50)(0.776) + (1/2)(-32.2)(0.776)²
Simplifying:
d ≈ 38.8 - 10.04737
d ≈ 28.75263 feet
Therefore, the height when the velocity is 25 ft/s is approximately 28.75 feet.