A person standing at the edge of a seaside cliff kicks a stone over the edge with a speed of vi = 14 m/s. The cliff is h = 39 m above the water's surface.
(a) How long does it take for the stone to fall to the water?
(b) With what speed does it strike the water?
(a)
d = Vi*t + (1/2)*a*t^2
Keep in mind that to use the equation to find time with respect to displacement in height, consider only the vertical component of the initial velocity. (You did not specify a direction for the initial velocity).
(b)
Having found the time from part (a), use a = (Vf - Vi)/t to find the velocity when it hits the water.
To solve this problem, we can use the equations of motion and the principles of free fall. We'll assume the stone is dropped vertically and neglect air resistance.
(a) To find the time it takes for the stone to fall to the water, we can use the equation of motion:
h = (1/2) * g * t^2
where h is the height of the cliff, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
Substituting the given values, we have:
39 m = (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation, we get:
t^2 = (2 * 39 m) / 9.8 m/s^2
t^2 = 78 m / 9.8 m/s^2
t^2 ≈ 7.959 seconds
Therefore, the time it takes for the stone to fall to the water is approximately 7.959 seconds.
(b) To find the speed of the stone when it strikes the water, we can use the equation for velocity in free fall:
v = g * t
where v is the final velocity of the stone (when it strikes the water), g is the acceleration due to gravity, and t is the time.
Substituting the given values, we have:
v = 9.8 m/s^2 * 7.959 seconds
v ≈ 77.2042 m/s
Therefore, the speed at which the stone strikes the water is approximately 77.2 m/s.
To find the time it takes for the stone to fall to the water (part a), we can use the equation of motion:
h = 1/2 * g * t^2
where:
h = height of the cliff = 39 m
g = acceleration due to gravity = 9.8 m/s^2 (assuming no air resistance)
t = time taken for the stone to fall
Rearranging the equation, we get:
t^2 = 2h / g
Substituting the given values, we have:
t^2 = 2 * 39 / 9.8
t^2 = 78 / 9.8
t^2 = 7.9591836734693878
Taking the square root of both sides, we find:
t ≈ 2.823 s (rounded to 3 decimal places)
So, it takes approximately 2.823 seconds for the stone to fall to the water.
To determine the speed at which the stone strikes the water (part b), we can use the equation of motion:
v = g * t
where:
v = velocity/speed of the stone
g = acceleration due to gravity = 9.8 m/s^2 (assuming no air resistance)
t = time taken for the stone to fall (found in part a)
Substituting the known values, we have:
v = 9.8 * 2.823
v ≈ 27.6374 m/s (rounded to 4 decimal places)
Therefore, the speed at which the stone strikes the water is approximately 27.6374 m/s.
a)2.910
b)11.089