A football is kicked at ground level with a speed of 18.0 m/s at an angle of 38.0 degrees to the horizontal. How much does it hit the ground?
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To determine how far the football hits the ground, we can use the equations of projectile motion.
First, we need to separate the initial velocity into horizontal and vertical components. The horizontal component remains unchanged throughout the motion, while the vertical component is affected by gravity.
The vertical component of the initial velocity (Viy) can be determined using trigonometry:
Viy = V * sin(angle)
Viy = 18.0 m/s * sin(38.0 degrees)
Viy ≈ 11.0 m/s
Next, we can calculate the time it takes for the football to hit the ground. Since the initial vertical velocity is in the upward direction, the time it takes for the football to reach its highest point (when the vertical velocity is zero) is the same as the time it takes to hit the ground.
Using the equation for vertical displacement, we have:
d = Vit + (1/2)at^2
Since the football hits the ground, the vertical displacement (d) is equal to zero. We can rearrange the equation as follows:
0 = Viy * t - (1/2) * g * t^2
Where g is the acceleration due to gravity (approximated as 9.8 m/s^2).
Simplifying the equation:
0 = (11.0 m/s) * t - (4.9 m/s^2) * t^2
We now have a quadratic equation. We can solve for t by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
For this equation:
a = -4.9 m/s^2
b = 11.0 m/s
c = 0
t = (-11.0 ± √(11.0^2 - 4 * -4.9 * 0)) / 2 * -4.9
After evaluating the equation, we find two possible solutions: t ≈ 0.0 s and t ≈ 2.24 s.
Since the time cannot be zero (as the football has already been kicked), the time it takes for the football to hit the ground is approximately 2.24 seconds.
Finally, we can use the horizontal component of the initial velocity to determine the distance the football travels before hitting the ground. The horizontal displacement (d_h) can be found using:
d_h = V * cos(angle) * t
Plugging in the values:
d_h = 18.0 m/s * cos(38.0 degrees) * 2.24 s
After evaluating the equation, we find that the football hits the ground approximately 25.4 meters away.