A football is kicked with an initial speed of 18 m/s at an angle of 36 degrees above the horizontal.
(a) What is the maximum height that it reaches?
(b) At what time does it land?
(c) How far horizontally does it travel?
To solve this problem, we can use the equations of motion for projectile motion. Projectile motion is the motion of an object thrown or propelled into the air, subject to gravity.
First, let's define the given information:
Initial speed (u) = 18 m/s
Angle (θ) = 36 degrees
Acceleration due to gravity (g) = 9.8 m/s^2 (considering a flat surface)
(a) To find the maximum height, we need to determine the vertical component of velocity (v_y) at the highest point. The initial vertical velocity (v_iy) can be calculated using the formula v_iy = u * sin(θ).
v_iy = 18 m/s * sin(36 degrees)
v_iy ≈ 18 * 0.5878
v_iy ≈ 10.5814 m/s
At the highest point, the final vertical velocity (v_fy) will be zero. We can use the equation v_fy^2 = v_iy^2 - 2 * g * h (where h is the maximum height) to find the maximum height.
0^2 = (10.5814 m/s)^2 - 2 * 9.8 m/s^2 * h
0 = 111.988m/s^2 - 19.6 m/s^2 * h
19.6 h = 111.988
h = 111.988 / 19.6
h ≈ 5.72 meters
Therefore, the maximum height the football reaches is approximately 5.72 meters.
(b) Now, let's find the time it takes for the football to land. The time of flight (T) can be calculated using the equation T = 2 * v_iy / g.
T = 2 * 10.5814 m/s / 9.8 m/s^2
T ≈ 2 * 1.0807 s
T ≈ 2.1614 s
Therefore, the time it takes for the football to land is approximately 2.1614 seconds.
(c) Finally, let's determine the horizontal distance traveled (range). The range (R) can be calculated using the equation R = u * cos(θ) * T.
R = 18 m/s * cos(36 degrees) * 2.1614 s
R ≈ 18 * 0.8090 * 2.1614
R ≈ 33.086 meters
Therefore, the football travels approximately 33.086 meters horizontally.
To solve this problem, we can break it down into three parts:
(a) Finding the maximum height:
To determine the maximum height, we need to find the vertical component of the football's initial velocity. We can use the equation for vertical displacement of an object in projectile motion:
y = (vi * sin(theta) * t) - (0.5 * g * t^2)
Where:
- y is the vertical displacement (in this case, the maximum height)
- vi is the initial velocity (18 m/s)
- theta is the angle of the initial velocity (36 degrees)
- t is the time taken to reach the maximum height
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since the football reaches its maximum height when its vertical velocity becomes zero, we can set the y-component of the velocity equal to zero and solve for t:
0 = (18 * sin(36) * t) - (0.5 * 9.8 * t^2)
Rearranging the equation, we get:
9.8t^2 = 18 * sin(36) * t
Dividing both sides by t and rearranging the equation, we get:
9.8t = 18 * sin(36)
Solving for t, we find:
t = (18 * sin(36)) / 9.8
Using a calculator to compute it:
t ≈ 1.422 seconds
Now that we have the time, we can substitute it back into the equation for y to find the maximum height:
y = (18 * sin(36) * 1.422) - (0.5 * 9.8 * (1.422)^2)
Using a calculator to compute it:
y ≈ 7.61 meters
So the football reaches a maximum height of approximately 7.61 meters.
(b) Finding the time of landing:
To find the time it takes for the football to land, we need to consider the total time of flight. The time of flight can be calculated using the equation:
T = (2 * vi * sin(theta)) / g
Where:
- T is the total time of flight
- vi is the initial velocity (18 m/s)
- theta is the angle of the initial velocity (36 degrees)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the values, we get:
T = (2 * 18 * sin(36)) / 9.8
Using a calculator to compute it:
T ≈ 3.71 seconds
Since the time of flight is the time it takes for the football to reach the ground, the time of landing is half of the time of flight:
Time of landing = 0.5 * T
Substituting the value of T we found:
Time of landing ≈ 0.5 * 3.71
Using a calculator to compute it:
Time of landing ≈ 1.855 seconds
So the football lands approximately 1.855 seconds after being kicked.
(c) Finding the horizontal distance traveled:
To find the horizontal distance traveled, we can use the equation for horizontal displacement of an object in projectile motion:
x = vi * cos(theta) * t
Where:
- x is the horizontal displacement
- vi is the initial velocity (18 m/s)
- theta is the angle of the initial velocity (36 degrees)
- t is the time of flight
Substituting the values, we get:
x = 18 * cos(36) * 3.71
Using a calculator to compute it:
x ≈ 52.85 meters
So the football travels approximately 52.85 meters horizontally.