A football is kicked off the ground at an initial upward velocity of 20 m/sec.
(We are using d=rt-5t^2)
a. Calculate the altitude after 2.7 sec
b. When will the ball be at 17 m above the ground?
c. When is the ball at the highest point?
d. How high is the ball at its highest point?
altitude= Vi*time- 4.9 time^2
use that for a, b.
For c, note that velocity at the top is zero.
Vtop=Vi*time-4.9t^2
solve for time
d. use the first equation again.
a.10
To solve these problems, we'll need to use the equation for the vertical distance of an object in freefall:
d = rt - 5t^2
where d is the distance, r is the initial velocity, t is time, and 5 is a constant.
a. To calculate the altitude after 2.7 seconds, we can substitute t = 2.7 seconds into the equation.
d = (20 * 2.7) - 5 * (2.7)^2
d = 54 - 5 * (7.29)
d = 54 - 36.45
d ≈ 17.55 meters
Therefore, the altitude after 2.7 seconds is approximately 17.55 meters.
b. To find when the ball will be at 17 meters above the ground, we need to set d (distance) equal to 17 and solve for t.
17 = 20t - 5t^2
5t^2 - 20t + 17 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 5, b = -20, and c = 17. Plugging these values into the formula, we get:
t = [20 ± sqrt((-20)^2 - 4 * 5 * 17)] / (2 * 5)
t = [20 ± sqrt(400 - 340)] / 10
t = [20 ± sqrt(60)] / 10
So the solutions are:
t1 = (20 + sqrt(60)) / 10
t2 = (20 - sqrt(60)) / 10
These represent the times when the ball is at 17 meters above the ground.
c. To find when the ball is at its highest point, we need to identify the time when the vertical velocity becomes zero. At the highest point, the vertical velocity flips from an upward value to a downward value.
The vertical velocity can be calculated using the equation:
v = r - 10t
Setting v = 0, we solve for t:
0 = 20 - 10t
10t = 20
t = 2 seconds
Therefore, the ball reaches its highest point at 2 seconds.
d. To find the height of the ball at its highest point, we can substitute t = 2 seconds into the equation for distance:
d = 20 * 2 - 5 * 2^2
d = 40 - 5 * 4
d = 40 - 20
d = 20 meters
Thus, the ball is at a height of 20 meters at its highest point.