an object is projected upward at a rate of 96 feet per second. find the time when it is 144 feet above the ground
h(t) = -16t^2 + 96t
144 = -16t^2 + 96t
t^2 - 6t + 9 = 0
(t-3)(t-3) = 0
t = 3
To find the time when the object is 144 feet above the ground, we can use the kinematic equation for motion under constant acceleration.
The formula to calculate the height (h) of an object in free fall at a given time (t) is:
h = initial height + (initial velocity x time) - (0.5 x acceleration x time^2).
In this case, the initial height is 0 feet (assuming the ground as the reference point), the initial velocity is 96 feet per second (projected upward), and the acceleration is the acceleration due to gravity, which is approximately -32 feet per second squared (assuming upward as positive). The negative sign indicates that the acceleration is directed downward.
Thus, the equation becomes:
144 = 0 + (96 x t) - (0.5 x (-32) x t^2).
Simplifying the equation, we have:
144 = 96t + 16t^2.
Moving all terms to one side, we get:
16t^2 + 96t - 144 = 0.
This is a quadratic equation that can be solved using various methods like factoring, completing the square, or using the quadratic formula. In this case, we can simply factor out 16:
16(t^2 + 6t - 9) = 0.
Dividing both sides by 16:
t^2 + 6t - 9 = 0.
Now we can solve this equation for t using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a),
where a = 1, b = 6, and c = -9.
Plugging in the values, we have:
t = (-6 ± √(6^2 - 4(1)(-9))) / (2(1)).
Simplifying further:
t = (-6 ± √(36 + 36)) / 2.
t = (-6 ± √72) / 2.
t = (-6 ± 6√2) / 2.
Now, we can simplify it as:
t = -3 ± 3√2.
So, the two possible values for time (t) are:
t = -3 + 3√2,
t = -3 - 3√2.
Since time cannot be negative in this context, we disregard the negative value.
Therefore, the time when the object is 144 feet above the ground is approximately -3 + 3√2 seconds.