If an object is thrown vertically upward with an initial velocity of v, from an original position of s, the height h at any time t is given by: h=-16t^2+vt+s(where h and s are in ft, t is in seconds and v is in ft/sec)
a ball is thrown upward with initial velocity of 96 ft/sec from the top of a 100 ft. bridge. Determine the time that it takes for the ball to get to a height of 200 ft.
Round your answers to 2 decimals, separated by a comma.
To determine the time it takes for the ball to reach a height of 200 ft, we need to solve the height equation h = -16t^2 + vt + s for the value of t when h = 200 ft.
Given:
Initial velocity, v = 96 ft/sec
Original position, s = 100 ft
Let's substitute these values into the height equation:
200 = -16t^2 + 96t + 100
To solve this quadratic equation, we can rearrange it into standard form:
16t^2 - 96t + 100 = 0
Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
For the equation 16t^2 - 96t + 100 = 0, the coefficients are:
a = 16, b = -96, c = 100
Plugging these values into the quadratic formula, we have:
t = (-(-96) ± √((-96)^2 - 4 * 16 * 100)) / (2 * 16)
Simplifying further:
t = (96 ± √(9216 - 6400)) / 32
t = (96 ± √2816) / 32
Now, using a calculator, we can find the approximate values of t:
t ≈ (96 ± 53.02) / 32
Calculating the two possible values of t:
t1 ≈ (96 + 53.02) / 32 ≈ 149.02 / 32 ≈ 4.65
t2 ≈ (96 - 53.02) / 32 ≈ 42.98 / 32 ≈ 1.34
Since time cannot be negative, we discard the negative value (t2) and keep the positive value (t1).
Therefore, the time it takes for the ball to reach a height of 200 ft is approximately 4.65 seconds.