a ball of mass 100g is dropped from the top of a vertical cliff 45m high. Given that the velocity just before striking the sandy beach is 30m/s and the ball penetrates the sand to a depth of 10cm. Determine its average retardation force
V^2 = Vo^2 + 2a*d = 0.
30^2 + 2a*0.1 = 0, 0.2a = -900, a = -4500 m/s^2.
F = M*a = 0.1 * (-4500)= -450 N.
To determine the average retardation force acting on the ball as it penetrates the sand, we can use the equations of motion.
First, let's calculate the initial velocity of the ball when it was dropped from the cliff. We can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity (30 m/s)
u = initial velocity (unknown)
a = acceleration due to gravity (-9.8 m/s^2, as the ball is falling downward)
s = distance traveled (45 m)
Rearranging the equation, we have:
u^2 = v^2 - 2as
Substituting the values into the equation:
u^2 = (30 m/s)^2 - 2(-9.8 m/s^2)(45 m)
u^2 = 900 m^2/s^2 + 882 m^2/s^2
u^2 = 1782 m^2/s^2
Taking the square root of both sides:
u = √1782 m/s
u ≈ 42.2 m/s
Now, let's calculate the deceleration of the ball as it penetrates the sand. We can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s, as the ball comes to rest after penetrating the sand)
u = initial velocity (30 m/s)
a = deceleration (unknown)
s = distance traveled in the sand (10 cm, which can be converted to 0.1 m)
Rearranging the equation, we have:
a = (v^2 - u^2) / (2s)
Substituting the values into the equation:
a = (0 m/s)^2 - (30 m/s)^2 / 2(0.1 m)
a = -900 m^2/s^2 / 0.2 m
a = -4500 m^2/s^2 / m
a = -4500 m/s^2
Now, we can calculate the average retardation force acting on the ball using Newton's second law of motion, which states that force (F) equals mass (m) multiplied by acceleration (a):
F = ma
Given the mass of the ball (100 g, which can be converted to 0.1 kg) and the deceleration (4500 m/s^2), we have:
F = (0.1 kg)(-4500 m/s^2)
F = -450 N
Therefore, the average retardation force acting on the ball is 450 N. Note that the negative sign indicates that the force is acting in the opposite direction to the motion of the ball.