A 2.63 kg ball is dropped from the roof of a building 180.6 m high. While the ball is falling to Earth, a horizontal wind exerts a constant force of 13.0 N on the ball. How long does it take to hit the ground? The acceleration of gravity is 9.81 m/s2 .
How far from the building does the ball hit the ground?
What is the speed when it hits the ground?
Vertical problem:
(1/2) m V^2 = m g h
V = sqrt (2 g h) at bottom
V =sqrt(2*9.81*180.6
time at ground:
average v = V/2
so
t = 180.6/v = time at ground
Horizontal problem (use t from above)
a = 13.0/2.63
u = a t
d = (1/2) a t^2
speed = sqrt(u^2+V^2)
To find the time it takes for the ball to hit the ground, we can use the kinematic equation:
h = (1/2)gt^2
Where:
- h is the height of the building (180.6 m)
- g is the acceleration due to gravity (9.81 m/s^2)
- t is the time it takes for the ball to hit the ground
Substituting the values into the equation:
180.6 = (1/2)(9.81)t^2
Solving for t:
t^2 = (2 * 180.6) / 9.81
t^2 = 36.79
t ≈ 6.07 s
Therefore, it takes approximately 6.07 seconds for the ball to hit the ground.
To find the horizontal distance from the building where the ball hits the ground, we can use the formula:
d = v * t
Where:
- d is the horizontal distance
- v is the horizontal velocity of the ball
- t is the time taken to hit the ground
Since there is a constant horizontal wind force acting on the ball, the horizontal velocity can be found using the equation:
F = m * a
Where:
- F is the horizontal force (13 N)
- m is the mass of the ball (2.63 kg)
- a is the horizontal acceleration
Since there is no vertical force acting on the ball, there is no change in vertical acceleration. Therefore, the horizontal acceleration is also 0.
Substituting the values into the equation:
13 = 2.63 * 0
As the value of a is 0, the horizontal force is also 0. Therefore, the horizontal velocity is 0.
Substituting the values of v = 0 and t = 6.07 into the earlier equation, we get:
d = 0 * 6.07
d = 0 meters
Therefore, the ball hits the ground directly below the building and the horizontal distance is 0 meters.
To find the speed at which the ball hits the ground, we can use the equation:
v = g * t
Substituting the values of g = 9.81 m/s^2 and t = 6.07 s, we get:
v = 9.81 * 6.07
v ≈ 59.52 m/s
Therefore, the speed at which the ball hits the ground is approximately 59.52 m/s.
To find out how long it takes for the ball to hit the ground, we can use the equation of motion:
d = v₀t + (1/2)at²
where:
d = distance traveled (180.6 m in this case)
v₀ = initial velocity (0 m/s since the ball is dropped, it does not have an initial horizontal velocity)
t = time taken (what we're trying to find)
a = acceleration due to gravity (-9.81 m/s² since it's directed downwards)
Substituting the known values, the equation becomes:
180.6 = 0 × t + (1/2)(-9.81)t²
Simplifying, we get:
180.6 = -4.905t²
Dividing both sides by -4.905, we have:
t² = -180.6 / -4.905
t² = 36.81
Taking the square root of both sides, we find:
t ≈ 6.07 seconds
So, it takes approximately 6.07 seconds for the ball to hit the ground.
To determine how far from the building the ball hits the ground horizontally, we use the equation:
d = vt
where:
d = distance traveled horizontally (what we're trying to find)
v = horizontal velocity (constant, provided by the horizontal wind force, which is 13.0 N in this case)
t = time taken (6.07 seconds)
Substituting the known values:
d = 13.0 × 6.07
d ≈ 78.91 meters
Therefore, the ball hits the ground approximately 78.91 meters away from the building.
To find the speed at which the ball hits the ground, we use the formula:
v = v₀ + at
where:
v = final velocity (what we're trying to find)
v₀ = initial velocity (0 m/s since the ball is dropped, it does not have an initial horizontal velocity)
a = acceleration due to gravity (-9.81 m/s² since it's directed downwards)
t = time taken (6.07 seconds)
Substituting the known values:
v = 0 + (-9.81) × 6.07
v ≈ -59.58 m/s
Since the velocity is negative, it indicates that the ball's direction is downwards. Therefore, the speed at which the ball hits the ground is approximately 59.58 m/s.