The height of a soccer ball kicked from the ground can be approximated by the function h = =16t^2 + 64t, where he is the height in feet and t is the time in seconds. Find the time it takes for the ball to return to the ground.
A: It takes the ball 2 seconds to return to the ground.
To find the time it takes for the ball to return to the ground, we need to find the value of t when the height of the ball, h, is equal to 0.
Given the function h = 16t^2 + 64t, we can set h to 0 and solve for t.
0 = 16t^2 + 64t
Now, we can factor out t:
0 = t(16t + 64)
To solve this equation, we have two options:
Option 1: Set t = 0
If we set t = 0, we get:
0 = 16(0) + 64(0)
0 = 0 + 0
This does not give us a valid solution because it means the ball has not yet been kicked.
Option 2: Set 16t + 64 = 0
If we set 16t + 64 = 0, we can solve for t:
16t + 64 = 0
Subtracting 64 from both sides:
16t = -64
Dividing both sides by 16:
t = -64/16
Simplifying:
t = -4
However, the time cannot be negative, so this solution is not valid.
Thus, the only valid solution is t = 0, which means the ball has not yet been kicked.
Therefore, it takes the ball 2 seconds to return to the ground.
To find the time it takes for the ball to return to the ground, we need to set the height (h) equal to zero (since the ball is on the ground when the height is zero) and solve for t.
So we have the equation:
h = 16t^2 + 64t
Setting h = 0, we get:
0 = 16t^2 + 64t
Now we can solve for t by factoring or using the quadratic formula.
Factoring:
0 = 16t(t + 4)
From this equation, we can see that t = 0 or t + 4 = 0.
t = 0 implies the ball is on the ground at the start, but we are interested in the time it takes for the ball to return to the ground after being kicked. So, we can ignore t = 0.
Now, solving t + 4 = 0, we get:
t = -4
Since time cannot be negative, we discard this solution as well.
Therefore, the only valid solution is t = 0. This means it takes the ball 2 seconds to return to the ground.
V = Vo + g*Tr = 0
Tr = (V-Vo)/g = (0-64)/-32 = 2 s. = Rise
time.
Tf = Tr = 2 s. = Fall time.
Tr+Tf = 2 + 2 = 4 s. = Time it takes to
return to gnd.