A 3.00 kg ball is dropped from the roof of a building 176.4 m high. While the ball is falling to Earth, a horizontal wind exerts a constant force of 12.0 N on it.
a.)How far from the building does the ball hit the ground?
b.)How long does it take to hit the ground?
c.)What is its speed when it hits the ground?
Let's solve the problem step-by-step:
Step 1: Determine the gravitational force on the ball.
The gravitational force on the ball is given by the formula:
F_gravity = mass * acceleration due to gravity
In this case, mass = 3.00 kg and acceleration due to gravity is approximately 9.8 m/s^2.
So, F_gravity = 3.00 kg * 9.8 m/s^2 = 29.4 N
Step 2: Find the net force acting on the ball.
The net force is the vector sum of all forces acting on the ball. In this case, it consists of the gravitational force and the horizontal wind force.
Net force = F_gravity + F_wind
Net force = 29.4 N + 12.0 N
Net force = 41.4 N
Step 3: Determine the time taken to hit the ground.
To find the time, we can use the kinematic equation:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
In this case, the initial velocity is 0 m/s because the ball is dropped.
The distance fallen is 176.4 m and the acceleration is the acceleration due to gravity (-9.8 m/s^2).
176.4 = (0 * time) + (0.5 * -9.8 * time^2)
Rearranging the equation, we get:
0.5 * -9.8 * time^2 = 176.4
Dividing both sides by -4.9:
time^2 = -36
Since time cannot be negative, we take the positive square root:
time = √36 = 6 s
The time taken for the ball to hit the ground is 6 seconds.
Step 4: Calculate the horizontal distance the ball travels.
The horizontal distance traveled can be calculated using the formula:
distance = velocity * time
In this case, the only force acting horizontally is the wind force, which is constant at 12.0 N. So the horizontal velocity is constant.
velocity = (force / mass) * time
velocity = (12.0 N) / (3.00 kg) * 6 s
velocity = 2 m/s
distance = velocity * time
distance = 2 m/s * 6 s
distance = 12 m
The ball hits the ground 12 meters from the building.
Step 5: Determine the speed of the ball when it hits the ground.
The final velocity of the ball when it hits the ground can be determined using the equation:
final velocity = initial velocity + acceleration * time
Since there is no initial velocity in the horizontal direction, the initial velocity is 0 m/s.
final velocity = 0 m/s + (-9.8 m/s^2) * 6 s
final velocity = -58.8 m/s
The speed of the ball when it hits the ground is 58.8 m/s. Note that the negative sign indicates the direction (downwards).
To find the answers to these questions, we can use the equations of motion and apply the basic principles of physics. Let's break down each question and explain how to solve them step by step.
a.) How far from the building does the ball hit the ground?
To find the horizontal distance the ball travels, we need to calculate the time it takes to hit the ground and then multiply it by the horizontal velocity of the ball.
First, we need to determine the initial velocity of the ball in the x-direction. Since no force acts in this direction, the horizontal speed remains constant. Therefore, the initial velocity in the x-direction is zero.
Next, we can calculate the time it takes to hit the ground using the equation:
h = (1/2)gt^2
Where:
h = height of the building (176.4 m)
g = acceleration due to gravity (9.8 m/s^2)
t = time
Rearranging the equation to solve for time (t), we get:
t = sqrt(2h / g)
Now, we can substitute the values into the equation to calculate the time:
t = sqrt(2 * 176.4 / 9.8)
Once we have the time, we can calculate the horizontal distance (d) using the formula:
d = v * t
Where:
v = horizontal velocity of the ball, which is zero in this case
Therefore, the answer to part a) is that the ball hits the ground at a distance of 0 meters from the building.
b.) How long does it take to hit the ground?
We have already calculated the time in the previous explanation. Substituting the values into the equation:
t = sqrt(2 * 176.4 / 9.8)
Calculating this, we find the time it takes for the ball to hit the ground.
c.) What is its speed when it hits the ground?
To find the speed of the ball when it hits the ground, we need to calculate the final velocity.
The final velocity (vf) can be calculated using the equation:
vf = sqrt(vi^2 + 2ad)
Where:
vi = initial velocity of the ball
a = acceleration due to gravity (9.8 m/s^2)
d = distance the ball falls (height of the building, 176.4 m)
Since there is no initial velocity in the y-direction (as the ball is dropped), vi would be zero. Therefore, the equation simplifies to:
vf = sqrt(2 * g * d)
Substituting the values into the equation, we can calculate the final velocity of the ball when it hits the ground.
By following these steps, we can find the answers to all the given questions.
a) F=ma or a=F/m
distancehorizontal=1/2 a t^2 where t is given by...
176.4=1/2 g t^2