The height of a ball thrown up with a velocity of 39.2 m/s is given as a function of time(measured in seconds) by the equation H(t) = 39.2t -4.9t2 + 58.8. How much time will the ball take to fall to the ground? (9.29 s)
pls explain how to solve the answer
H(finalTime)=0=39.2t -4.9t2 + 58.8
solve that quadratic.
To find the time it takes for the ball to fall to the ground, we need to determine when the height (H) is equal to zero.
The equation for the height of the ball at any given time (t) is given as H(t) = 39.2t - 4.9t^2 + 58.8. Since we want to find when the ball hits the ground, we set H(t) equal to zero and solve for t.
0 = 39.2t - 4.9t^2 + 58.8
Next, we rearrange the equation and rewrite it in standard quadratic form:
4.9t^2 - 39.2t + 58.8 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we will use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Where a, b, and c are the coefficients of the quadratic equation. In this case, a = 4.9, b = -39.2, and c = 58.8.
t = (-(-39.2) ± √((-39.2)^2 - 4 * 4.9 * 58.8)) / (2 * 4.9)
Simplifying the equation further:
t = (39.2 ± √(1536.64 - 1147.04)) / 9.8
t = (39.2 ± √(389.6)) / 9.8
Calculating the square root:
t = (39.2 ± 19.739) / 9.8
To find the two possible values of t, we calculate for both the positive and negative square root:
t1 = (39.2 + 19.739) / 9.8 = 5.59 seconds
t2 = (39.2 - 19.739) / 9.8 = 1.86 seconds
Since the ball is initially thrown upwards, we are interested in the time it takes for it to reach the maximum height and fall back down. Therefore, we discard the value of t1 (5.59 seconds) as it corresponds to the time it takes the ball to reach the maximum height.
The time it takes for the ball to fall back to the ground is approximately t2 = 1.86 seconds.
To find the time it takes for the ball to fall to the ground, we need to determine when the height of the ball is zero. In other words, we need to solve the equation H(t) = 0.
Given H(t) = 39.2t - 4.9t^2 + 58.8, we can substitute H(t) with 0 and solve for t:
0 = 39.2t - 4.9t^2 + 58.8
Rearranging the equation:
4.9t^2 - 39.2t + 58.8 = 0
Now, we can use the quadratic formula to solve for t. The quadratic formula states:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 4.9t^2 - 39.2t + 58.8 = 0, we have a = 4.9, b = -39.2, and c = 58.8.
Using the quadratic formula with these values:
t = (-(-39.2) ± √((-39.2)^2 - 4 * 4.9 * 58.8)) / (2 * 4.9)
Simplifying:
t = (39.2 ± √(1536.64 - 1151.04)) / 9.8
t = (39.2 ± √385.6) / 9.8
Calculating the square root:
t = (39.2 ± 19.64) / 9.8
Now, let's calculate the two possible values for t:
1. t = (39.2 + 19.64) / 9.8
t = 58.84 / 9.8
t ≈ 6 seconds
2. t = (39.2 - 19.64) / 9.8
t = 19.56 / 9.8
t ≈ 2 seconds
Since we are considering the time it takes for the ball to fall to the ground, we discard the solution t = 2 seconds. Therefore, the ball will take approximately 6 seconds to fall to the ground.