A bullet is fired straight up from a gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula h=−16t2+v0t+8 to determine how many seconds it will take for the bullet to hit the ground. (That is, when will h=0?) Round your answer to 2 decimal places.
0 = -16 t^2 + 1120 t + 8
0 = -2 t^2 + 140 t + 1
use the quadratic formula to find t
To determine how many seconds it will take for the bullet to hit the ground, we can set the equation h = 0 and solve for t. The given formula is:
h = -16t^2 + v0t + 8
Setting h = 0, the equation becomes:
0 = -16t^2 + 1120t + 8
We can now solve this quadratic equation. To do so, we can use either factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula since it works for any quadratic equation.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -16, b = 1120, and c = 8. Substituting these values into the quadratic formula, we get:
t = (-1120 ± √(1120^2 - 4(-16)(8))) / (2(-16))
Simplifying further:
t = (-1120 ± √(1254400 + 512)) / (-32)
t = (-1120 ± √(1254912)) / (-32)
t = (-1120 ± 1119.99) / (-32)
Now, we can solve for t:
t = (-1120 + 1119.99) / (-32) or t = (-1120 - 1119.99) / (-32)
t = -0.00313 or t = 70
We ignore the negative value since time cannot be negative in this context.
Therefore, it will take approximately 70 seconds for the bullet to hit the ground (when h=0).
To determine how many seconds it will take for the bullet to hit the ground, we can use the given formula h = -16t^2 + v0t + 8, where h represents the height of the bullet at time t and v0 is the initial velocity of the bullet.
We need to find the value of t when h = 0, because when the bullet hits the ground, its height will be zero.
Substituting h = 0 into the formula, we get:
0 = -16t^2 + 1120t + 8
This equation is a quadratic equation, and we need to solve for t. To do this, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation: a = -16, b = 1120, and c = 8.
Substituting these values into the quadratic formula, we have:
t = (-1120 ± √(1120^2 - 4 * -16 * 8)) / (2 * -16)
Now we can calculate the value of t by solving this equation.
t = (-1120 ± √(1254400 + 512)) / (-32)
t = (-1120 ± √1254912) / (-32)
Calculating the square root, we get:
t = (-1120 ± 1119.84) / (-32)
Dividing both numerator and denominator by -32, we have:
t = (1120 ± 1119.84) / 32
Now we have two possible values for t:
t1 = (1120 + 1119.84) / 32 ≈ 70.37
t2 = (1120 - 1119.84) / 32 ≈ 0.01
Since the bullet was initially fired upwards, it first reaches its maximum height and then falls back down to the ground. Therefore, the positive value t1 = 70.37 represents the time it takes for the bullet to hit the ground.
Therefore, it will take approximately 70.37 seconds for the bullet to hit the ground.