A football on the moon (1/6 earth's gravity) is kicked straight up with an initial velocity of +14.5 m/s. If it starts 2.7 m above the surface, how much time has passed until it is 16.8 m above the surface and on its way down?
16.8 = 2.7 + 14.5 t -(1/12)9.8 t^2
or
14.1 = 14.5 t - .817 t^2
t^2 -14.5 t + 14.1 = 0
t = [ 14.5 +/- sqrt (14.5^2 -4*14.1) ]/2
t = [ 14.5 +/- 12.4 ] / 2
= 1.05 or 13.5
the 1.05 is on the way up
so the answer is 13.5 seconds
To solve this problem, we can use the equations of motion to calculate the time it takes for the football to reach a height of 16.8 m above the surface of the moon.
First, we need to determine the direction of the initial velocity. The velocity of +14.5 m/s indicates that the football was kicked upwards, which will be the positive direction.
We can use the following equation of motion to solve for time:
Δy = v0t + (1/2)at^2,
where Δy represents the change in position, v0 is the initial velocity, t is the time, and a is the acceleration.
Given:
v0 = +14.5 m/s (initial velocity)
Δy = 16.8 m - 2.7 m = 14.1 m (change in position)
g = 1/6(9.8 m/s^2) = 1.633 m/s^2 (acceleration due to gravity on the moon)
Plugging in the values into the equation of motion:
14.1 m = (+14.5 m/s)t + (1/2)(1.633 m/s^2)t^2.
This equation is a quadratic equation in the form of at^2 + bt + c = 0, where a = (1/2)(1.633 m/s^2), b = +14.5 m/s, and c = -14.1 m.
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values:
t = [-(+14.5) ± √((+14.5)^2 - 4(1/2)(1.633)(-14.1))] / (2(1/2)(1.633)).
Simplifying the expression:
t = (-14.5 ± √(210.25 + 91.17852)) / (1.633).
Calculating the values under the square root:
t ≈ (-14.5 ± √301.42852) / (1.633).
Now, calculate the time:
t ≈ (-14.5 ± 17.36) / (1.633).
Solving both possible solutions:
t ≈ (-14.5 + 17.36) / (1.633) ≈ 1.764 s,
t ≈ (-14.5 - 17.36) / (1.633) ≈ -3.808 s.
Since time cannot be negative, we discard the negative value. Therefore, the time it takes for the football to reach a height of 16.8 m above the surface and on its way down is approximately 1.764 seconds.