The height , in metres, of the trajectory of a football is given by h(t)=2=28t-4.9t^2, where t is the time in height of the football and the time when the height is reached
To find the time when the height is reached, we can set h(t) equal to the desired height and then solve for t.
Given that h(t) = 28t - 4.9t^2, let's say we want to find the time when the height is 2 meters. We substitute h(t) with 2 and solve the resulting equation for t:
2 = 28t - 4.9t^2
To make the equation easier to work with, let's rearrange it into the standard quadratic form:
4.9t^2 - 28t + 2 = 0
Now we have a quadratic equation in the form of at^2 + bt + c = 0, where a = 4.9, b = -28, and c = 2. To find the values of t that satisfy this equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c from our equation:
t = (-(-28) ± √((-28)^2 - 4(4.9)(2))) / (2(4.9))
Simplifying further:
t = (28 ± √(784 - 39.2)) / 9.8
t = (28 ± √(744.8)) / 9.8
t ≈ (28 ± 27.26) / 9.8
This gives us two possible values for t:
t1 ≈ (28 + 27.26) / 9.8 ≈ 5.7 seconds
t2 ≈ (28 - 27.26) / 9.8 ≈ 0.8 seconds
Therefore, the football reaches a height of 2 meters at approximately t = 5.7 seconds and t = 0.8 seconds.
To find the time when the football reaches a certain height, we can solve the given equation.
The equation is h(t) = 28t - 4.9t^2.
Let's assume the height at which you want to find the time is h. So, we have:
h(t) = h.
Substituting this in the equation, we get:
h = 28t - 4.9t^2.
Rearranging the equation, we have a quadratic equation in terms of t:
4.9t^2 - 28t + h = 0.
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 4.9, b = -28, and c = h.
Substituting these values into the quadratic formula, we get:
t = (-(-28) ± √((-28)^2 - 4 * 4.9 * h)) / (2 * 4.9).
Simplifying further:
t = (28 ± √(784 - 19.6h)) / 9.8.
This formula gives us two possible values for t, one with the positive sign (+) and another with the negative sign (-). Take note that the negative value of t may not be physically meaningful in this context.
So, to find the time when the football reaches a certain height, substitute the desired height (h) into the formula:
t = (28 ± √(784 - 19.6h)) / 9.8.
Solve for t using this formula, and you will find the time(s) when the football reaches the specified height.