You throw a tennis ball up in the air from an initial height of 4 feet and an initial velocity of 36 feet per
second. Will the ball reach a height of 30 feet? If so, how long will it take?
you should know that
height = -16t^2 + 36t + 4
so 30 = -16t^2 + 36t + 4
16t^2 - 36t + 26 = 0
8t^2 - 18t + 13 = 0
use the quadratic equation to solve for t
( you should get an imaginary root, so the ball never reaches a height of 30 feet)
To determine whether the ball will reach a height of 30 feet and how long it will take, we can use the equations of motion. Specifically, we can use the kinematic equation that relates displacement, initial velocity, time, and acceleration.
The kinematic equation we need to use is:
s = ut + (1/2)at^2
Where:
s is the displacement (change in height)
u is the initial velocity
t is the time
a is the acceleration due to gravity (approximately -32.2 ft/s^2 for objects thrown upwards near the Earth's surface)
In this case, the initial height is given as 4 feet and the initial velocity is given as 36 feet per second. We want to find the time it takes for the ball to reach a height of 30 feet, so we need to find the time when the displacement is 30 - 4 = 26 feet.
Using the kinematic equation, we can plug in the known values:
26 = 36t + (1/2)(-32.2)t^2
This equation is a quadratic equation in terms of t. We can solve it by rearranging it to the form:
(1/2)(-32.2)t^2 + 36t - 26 = 0
Now we can solve this quadratic equation using various methods. One approach is to use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = (1/2)(-32.2), b = 36, and c = -26. Plugging in these values, we can calculate the time it takes for the ball to reach a height of 30 feet.
t = (-36 ± √(36^2 - 4(1/2)(-32.2)(-26))) / (2(1/2)(-32.2))
By evaluating this expression, we can find the values of t. Remember to take into account whether the solution is physically meaningful (i.e., positive values representing time).
So, to summarize, to determine if the ball reaches a height of 30 feet and the time it takes, we need to solve the quadratic equation (1/2)(-32.2)t^2 + 36t - 26 = 0 using the quadratic formula.