A place kicker kicks a football at an angle of 42 degrees above the ground with a speed of 22 m/s. A) find time to max. height? B) find time to the ground? C) find max. height of the ball? D) find the horizontal displacement of the kickoff?
To find the solutions to these questions, we can use the kinematic equations of motion. Let's break down each part:
A) To find the time to reach maximum height, we need to determine the vertical component of the initial velocity. We can use the equation:
v(vertical) = v(initial) * sin(angle)
Given that the initial velocity is 22 m/s and the angle is 42 degrees, we can calculate the vertical component of the initial velocity:
v(vertical) = 22 m/s * sin(42 degrees)
Solving for v(vertical):
v(vertical) ≈ 14.8 m/s
Now, we can use the equation for time of flight to reach maximum height:
t(max height) = v(vertical) / g
Since the acceleration due to gravity (g) on Earth is approximately 9.8 m/s^2:
t(max height) ≈ 14.8 m/s / 9.8 m/s^2
Calculating t(max height):
t(max height) ≈ 1.51 seconds
B) To find the time to reach the ground, we know that the time of flight from the ground to maximum height is equal to the time of flight from maximum height to the ground. Thus, the total time of flight is:
t(total) = 2 * t(max height)
Calculating t(total):
t(total) ≈ 2 * 1.51 seconds
t(total) ≈ 3.03 seconds
C) To find the maximum height of the ball, we can use the equation for vertical displacement:
h(max) = (v(vertical))^2 / (2 * g)
Substituting the known values:
h(max) ≈ (14.8 m/s)^2 / (2 * 9.8 m/s^2)
Calculating h(max):
h(max) ≈ 11.31 meters
D) To find the horizontal displacement of the kickoff, we need to calculate the horizontal component of the initial velocity. We can use the equation:
v(horizontal) = v(initial) * cos(angle)
Given that the initial velocity is 22 m/s and the angle is 42 degrees, we can calculate the horizontal component of the initial velocity:
v(horizontal) = 22 m/s * cos(42 degrees)
Solving for v(horizontal):
v(horizontal) ≈ 16.1 m/s
Since the horizontal velocity is constant, we can use the equation for horizontal displacement:
d(horizontal) = v(horizontal) * t(total)
Substituting the known values:
d(horizontal) ≈ 16.1 m/s * 3.03 seconds
Calculating d(horizontal):
d(horizontal) ≈ 48.78 meters
Therefore, the answers are:
A) The time to reach maximum height is approximately 1.51 seconds.
B) The time to reach the ground is approximately 3.03 seconds.
C) The maximum height of the ball is approximately 11.31 meters.
D) The horizontal displacement of the kickoff is approximately 48.78 meters.