How Tall is the Building the Child is Standing On?
Solving for the Height of the Building
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 37° from the edge of the building with an initial velocity of 21 m/s and lands 63 meters away from the wall. How tall, in meters, is the building that the child is standing on?
Answer: h = 21.5 m
Explanation:
First of all, we define a pair of coordinate axes along the horizontal and vertical direction, calling x-axis the horizontal and y-axis the vertical, with the origin at the point where the ball is kicked.
Neglecting air resistance, the only influence on the ball once kicked is due to gravity, so the ball is accelerated by the Earth with a constant value of -9.8 m/s2 (assuming the upward direction as positive).
So, we can use the kinematic equation for displacement for the vertical direction, as follows:
Δy = voy* t - 1/2*g*t2 (1)
Since the ball is kicked at an angle of 37º from the edge of the building, at an initial velocity of 21 m/s, we can find the horizontal and vertical initial speeds as follows:
vox = v* cos 37 = 21 m/s * cos 37 = 16.8 m/s (2)
voy = v* sin 37 = 21 m/s * sin 37 = 12.6 m/s (3)
In the horizontal direction, since gravity has no component in this direction, the ball moves at a constant speed, equal to v₀ₓ.
Applying the definition of average velocity, since we know the horizontal distance traveled, we can find the total time that the ball was in the air, as follows:
t = Δx/vox = 63m / 16.8m/s = 3.75 s (4)
Replacing (4) and (3) in (1), we can find the total vertical displacement, which is equal to the height of the building, as follows:
-h = 12.6m/s*3.75s -1/2*(9.8m/s2)*(3.75s)2 = -21.5 m (5)
Therefore, the height of the building that the child is standing on is h = 21.5 meters.
How did we determine the height of the building in this scenario?
We determined the height of the building by analyzing the kinematics of the soccer ball's motion after being kicked by the child on the building's roof. By considering the vertical displacement, initial velocities, time of flight, and the effects of gravity, we were able to calculate the height of the building to be 21.5 meters.