I know this has been posted many times but I keep trying and cannot get the right answer, A place kicker must kick a football from a point 39.9 m from a goal. As a result of the kick, the ball must clear the crossbar, which is 3.05 m high. When kicked the ball leaves the ground with a speed of 20.6 m/s at an angle of 53° to the horizontal. By how much does the ball clear or fall short of clearing the crossbar?
I got that the time was 3.35 and because v=d/t I got 11.91 and then made a triangle of 20.6 aas the hyp and 11.91 at the bottom so then i got 16.81 which crosses the bar by 13.75 but that's not the right answer, PLEASE HELP
To solve this problem, we can break it down into two components: the vertical and horizontal motion of the ball.
First, let's find the time it takes for the ball to reach its maximum height. We can use the vertical motion equation:
y = y0 + v0y * t - (1/2) * g * t^2
Where:
y = final vertical displacement (height of the crossbar) = 3.05 m
y0 = initial vertical displacement (initial height of the ball) = 0 m
v0y = vertical component of initial velocity = v0 * sin(53°)
g = acceleration due to gravity = 9.8 m/s^2
t = time taken to reach the maximum height (unknown)
Plugging in the values:
3.05 = 0 + (20.6 * sin(53°)) * t - (1/2) * 9.8 * t^2
Simplifying the equation:
9.8 * t^2 - (20.6 * sin(53°)) * t + 3.05 = 0
Solving this quadratic equation, we find that t ≈ 1.46 seconds.
Now, let's calculate the vertical displacement of the ball at this time:
y = y0 + v0y * t - (1/2) * g * t^2
y = 0 + (20.6 * sin(53°)) * 1.46 - (1/2) * 9.8 * (1.46)^2
y ≈ 16.81 m
We can see that your calculation for the vertical displacement is correct.
Next, let's find the horizontal displacement of the ball at the same time. We can use the horizontal motion equation:
x = x0 + v0x * t
Where:
x = horizontal displacement = 39.9 m
x0 = initial horizontal displacement = 0 m
v0x = horizontal component of initial velocity = v0 * cos(53°)
t = time taken to reach the maximum height (1.46 seconds)
Plugging in the values:
39.9 = 0 + (20.6 * cos(53°)) * 1.46
Simplifying the equation:
39.9 ≈ (20.6 * cos(53°)) * 1.46
Solving for (20.6 * cos(53°)):
(20.6 * cos(53°)) ≈ 11.303
Plugging this value back into the equation:
39.9 ≈ 11.303 * 1.46
Solving for 39.9:
39.9 ≈ 16.493
So the ball travels approximately 16.493 m horizontally.
Now, let's find how much the ball clears or falls short of clearing the crossbar. We can subtract the vertical displacement at the maximum height from the height of the crossbar:
Clearance = 16.81 m - 3.05 m ≈ 13.76 m
Therefore, the ball clears the crossbar by approximately 13.76 m.
I hope this explanation helps you understand how to solve the problem.