A 1.4 kg ball is thrown upwards off the side of a building with a spped of 5.8 m/s and falls into an ocean 16 meters below. Once the ball hits the water it travels a depth of 1.4m then floats to the surface. Assuming the acceleration of the ball when it hits the water is constant calculate the total time it takes from the time the ball is thrown to the time it "pops" out of the water. What is the average force the water exerts on the ball while it is sinking?
To calculate the total time it takes for the ball to reach the water's surface, we need to follow these steps:
1. Calculate the time taken for the ball to reach the highest point (where its vertical velocity becomes zero) during its upward motion using the given initial velocity and the gravitational acceleration.
2. Calculate the time taken for the ball to fall from the highest point to the water's surface using the vertical distance and gravitational acceleration.
3. Add the time taken in steps 1 and 2 to get the total time.
Step 1: Calculating the time taken to reach the highest point
To calculate the time taken for the ball to reach the highest point, we can use the kinematic equation:
v = u + at,
where:
v = final velocity (which is zero at the highest point),
u = initial velocity (5.8 m/s),
a = acceleration (the negative value of gravitational acceleration since the ball is moving upward),
t = time.
Rearranging the equation, we get:
t = (v - u) / a.
Substituting the values into the equation:
t1 = (0 - 5.8) / (-9.8),
t1 = 0.5918 seconds (approx.).
Step 2: Calculating the time taken to fall to the water's surface
To calculate the time taken for the ball to fall from the highest point to the water's surface, we can use the kinematic equation:
s = ut + (1/2)at^2,
where:
s = vertical distance (16m),
u = initial velocity (zero since the ball's vertical velocity is zero at the highest point),
a = acceleration (gravitational acceleration),
t = time.
Rearranging the equation, we get:
t = sqrt((2s) / a).
Substituting the values into the equation:
t2 = sqrt((2 * 16) / 9.8),
t2 = 2.02 seconds (approx.).
Step 3: Calculating the total time taken
To calculate the total time taken, we add t1 and t2:
total time = t1 + t2,
total time = 0.5918 + 2.02,
total time = 2.61 seconds (approx.).
Now, let's calculate the average force exerted by the water on the ball while it is sinking.
Average force can be calculated using Newton's second law of motion:
F = m * a,
where:
F = force,
m = mass (1.4 kg),
a = acceleration.
The acceleration is determined by the change in velocity divided by the time taken:
a = (final velocity - initial velocity) / time taken,
here, the final velocity is zero (the ball "pops" out of the water and its velocity is zero).
Hence,
a = -u / total time,
a = -5.8 / 2.61,
a = -2.222 m/s^2 (approx.).
Now, substitute the values into Newton's second law:
F = 1.4 * (-2.222),
F = -3.11 N (approx.).
The average force exerted by the water on the ball while it is sinking is approximately -3.11 N. The negative sign indicates that the direction of force is opposite to the direction of motion.