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 41° from the edge of the building with an initial velocity of 21 m/s and lands 59 meters away from the wall. How tall is the building that the child is standing on?
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To determine the height of the building, we can use kinematic equations of motion.
Let's break down the given information:
- The initial velocity of the soccer ball is 21 m/s.
- The angle of the kick from the edge of the building is 41°.
- The horizontal distance the soccer ball lands away from the wall is 59 meters.
First, let's find the time it takes for the soccer ball to hit the ground. Since the vertical and horizontal motions are independent, we can ignore the horizontal velocity (since it remains constant) and focus on the vertical motion.
The equation we can use to find the time of flight is:
t = (2 * v * sinθ) / g
Where:
t is the time of flight
v is the initial velocity (21 m/s)
θ is the angle of the kick (41°)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Let's calculate the time of flight:
t = (2 * 21 * sin(41°)) / 9.8
Using a calculator, the value of sin(41°) is approximately 0.65605902899. Substitute this value:
t = (2 * 21 * 0.65605902899) / 9.8
t ≈ 2.09 seconds
Now that we have the time of flight, we can determine the height of the building using the vertical displacement formula:
y = v * t * sinθ - 0.5 * g * t^2
Where:
y is the vertical displacement (height of the building)
v is the initial velocity (21 m/s)
t is the time of flight (2.09 seconds)
θ is the angle of the kick (41°)
g is the acceleration due to gravity (9.8 m/s^2)
Let's calculate the height of the building:
y = 21 * 2.09 * sin(41°) - 0.5 * 9.8 * 2.09^2
Using a calculator, the value of sin(41°) is approximately 0.65605902899, and 2.09^2 is approximately 4.3681. Substitute these values:
y = 21 * 2.09 * 0.65605902899 - 0.5 * 9.8 * 4.3681
y ≈ 17.45 - 21.48
y ≈ -4.03 meters
Based on the calculations, the height of the building is approximately 4.03 meters. However, since the result is negative, it suggests that the child is actually at a lower elevation than the reference point (the ground).
Vo = 21m/s[41o]
Xo = 21*c0s41 = 15.85 m/s.
Yo = 21*sin41 = 13.78 m/s.
D = Xo * Tf = 59 m.
15.85 * Tf = 59
Tf = 3.722 s. = Fall time.
h = Yo*t + 0.5g*t^2
h = 13.78*3.722 + 4.9*3.722^2 = 119.2 m