Determining the Height of a Building using Projectile Motion
You are walking around your neighborhood and you see a child on top of a roof of a building kick a soccer ball.
The soccer ball is kicked at 51° from the edge of the building with an initial velocity of 13 m/s and lands 63 meters away from the wall. How tall, in meters, is the building that the child is standing on?
Question:
How tall is the building that the child is standing on?
Answer:
Height of the building: 213 meters
Explanation: From the exercise we know that the child hits the ball with an initial velocity, its direction and where it hits the ground.
First of all, we need to calculate how long does it take to hit the ground:
63 meters = (13cos(51) m/s) t
t = 63m / 13cos(51) m/s = 7.70s
Now, from free falling object's formula, we know that position is:
0 = y_{o} + (13sin(51) m/s) * 7.70s - 0.5 * 9.8m/s^2 * (7.70s)^2
y_{o} = (4.9 m/s^2) * (7.70s)^2 - (13sin(51) m/s) * 7.70s = 213 meters
So, the building is 213 meters tall.
Final answer: To determine the height of the building, we can use the principles of projectile motion. First, we find the initial vertical velocity of the ball. Then, we calculate the time it takes for the ball to hit the ground. Finally, we subtract the distance the ball traveled from the ground level to find the height of the building.
Explanation: To determine the height of the building, we can use the principles of projectile motion. The horizontal component of the ball's motion does not affect its vertical height, so we can focus on the vertical component.
We know the initial velocity of the ball (13 m/s) and the angle at which it was kicked (51°). Using trigonometry, we can find the initial vertical velocity of the ball.
Next, we can use the equations of motion to find the time it takes for the ball to hit the ground. Since the ball lands 63 meters away from the wall, we can use this distance and the time to calculate the vertical distance the ball traveled.
Finally, we can calculate the height of the building by subtracting the distance the ball traveled from the ground level to the vertical distance the ball traveled.