A kicker punts a football from 3 feet above the ground with an initial velocity of 47 feet per second.
Write an equation that gives the height (in feet) of the football as a function of the time (in seconds) since it was punted.
Find the height (in feet) of the football 2 seconds after the punt.
Calculate how many seconds after the punt the ball would hit the ground.
see http://www.jiskha.com/display.cgi?id=1355105772
To write the equation, we start with the equation of motion for an object in free fall:
h(t) = h0 + v0t - (1/2)gt^2
where:
h(t) is the height of the object at time t
h0 is the initial height (in this case 3 feet)
v0 is the initial velocity (in this case 47 feet per second)
g is the acceleration due to gravity (approximately 32.2 feet per second squared)
t is the time since the object was punted.
In this case, the kicker punted the football from a height of 3 feet, so h0=3, and the initial velocity is 47 feet per second, so v0=47. Plugging in these values, we have:
h(t) = 3 + 47t - (1/2)(32.2)t^2
Now let's find the height of the football 2 seconds after the punt. We can substitute t=2 into the equation:
h(2) = 3 + 47(2) - (1/2)(32.2)(2)^2
h(2) = 3 + 94 - 64.4
h(2) = 32.6 feet
So, the height of the football 2 seconds after the punt is 32.6 feet.
To find how many seconds it will take for the ball to hit the ground, we set h(t) = 0, since the ball hits the ground when the height is 0. The equation becomes:
0 = 3 + 47t - (1/2)(32.2)t^2
Rearranging, we get:
(1/2)(32.2)t^2 - 47t - 3 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = [-b ± sqrt(b^2 - 4ac)] / (2a)
where a = (1/2)(32.2), b = -47, and c = -3. Plugging in these values:
t = [-(-47) ± sqrt((-47)^2 - 4(1/2)(32.2)(-3))] / (2(1/2)(32.2))
Simplifying further:
t = [47 ± sqrt(47^2 + 4(1/2)(32.2)(3))] / (32.2)
t = [47 ± sqrt(2209 + 193.2)] / 32.2
t = [47 ± sqrt(2402.2)] / 32.2
Since we're looking for the time it takes for the ball to hit the ground, we can disregard the negative solution. Therefore:
t = (47 + sqrt(2402.2)) / 32.2
Using a calculator, we find:
t ≈ 3.656 seconds
So, it will take approximately 3.656 seconds for the ball to hit the ground.