A 3.60 kg ball is dropped from the roof of a building 184.8 m high. While the ball is falling to earth, a horizontal wind exerts a constant force of 12.1 N on the ball.
How long does it take to hit the ground?
(acceleration of gravity = 9.81 m/s^2)
The horizontal force does not affect the vertical equation of motion. (However, wind force and the object's mass do affect where the ball lands, horizontally). The mass does not affect vertical motion, either. The vertical distance travelled is
Y = (1/2) g t^2
Set Y = 184.8 m and solve for t.
To find the time it takes for the ball to hit the ground, we can use the equations of motion. First, we need to determine the vertical (downward) acceleration of the ball.
In this case, the only vertical force acting on the ball is its weight, which is given by:
Weight = mass x acceleration due to gravity
Weight = (3.60 kg) x (9.81 m/s^2)
Weight = 35.316 N
Since the ball is subject to a horizontal force of 12.1 N, the only vertical force acting on the ball is its weight minus the force due to the wind resistance. Therefore, the net force in the downward direction is:
Net force in the downward direction = Weight - Force due to wind resistance
Net force in the downward direction = 35.316 N - 12.1 N
Net force in the downward direction = 23.216 N
Using Newton's second law of motion, we can relate this net force to the downward acceleration of the ball:
Net force = mass x acceleration
23.216 N = (3.60 kg) x acceleration
acceleration = 23.216 N / 3.60 kg
acceleration = 6.449 m/s^2
Now, we can determine the time it takes for the ball to hit the ground. We can use the equation for displacement in terms of initial velocity, time, and acceleration:
Displacement = (initial velocity) x time + (1/2) x acceleration x time^2
Since the ball is dropped from rest, the initial velocity is 0. The displacement is equal to the height of the building: 184.8 m. By substituting these values into the equation, we can solve for the time:
184.8 m = (0) x time + (1/2) x (6.449 m/s^2) x time^2
Rearranging the equation, we have a quadratic equation:
(1/2) x (6.449 m/s^2) x time^2 = 184.8 m
Multiplying both sides by 2 to eliminate the fraction, we get:
(6.449 m/s^2) x time^2 = 369.6 m
Dividing both sides by (6.449 m/s^2), we have:
time^2 = 57.262 s^2
Taking the square root of both sides, we find:
time = √(57.262 s^2)
time ≈ 7.57 seconds (rounded to two decimal places)
Therefore, it takes approximately 7.57 seconds for the ball to hit the ground.