The height of a projectile fired upward is given by the formula
s = v0t − 16t2,
where s is the height, v0 is the initial velocity, and t is the time. Find the time for a projectile to return to Earth if it has an initial velocity of 184 ft/s.
Different initial up speed but use :
if Sam fires a bullet straight upward with an initial velocity of 128 ft/sec, the height h of the bullet above the ground t sec after it is fired can be modeled by the equation h(t)=-16(t-4)^2+259
To find the time for the projectile to return to Earth, we need to find the value of t when the height, s, is equal to zero.
The formula for the height of the projectile is given as:
s = v0t - 16t^2
If the projectile returns to Earth, its height will be zero, so we can set s = 0 and solve for t.
0 = v0t - 16t^2
Rearranging the equation, we have:
16t^2 = v0t
Divide both sides of the equation by t:
16t = v0
Now, substitute the given value for the initial velocity, v0 = 184 ft/s:
16t = 184
Divide both sides of the equation by 16:
t = 184/16
Simplifying the right side of the equation:
t = 11.5
Therefore, the time for the projectile to return to Earth is 11.5 seconds.
To find the time for a projectile to return to Earth, we need to solve the equation s = v0t - 16t^2 for t, where s is the height and v0 is the initial velocity.
In this case, the projectile is returning to Earth, so the height will be 0. Therefore, we can rewrite the equation as:
0 = v0t - 16t^2
Let's plug in the given initial velocity, v0 = 184 ft/s:
0 = 184t - 16t^2
This is a quadratic equation in terms of t. We can now use the quadratic formula to solve for t. The quadratic formula states:
t = (-b ± √(b^2 - 4ac)) / (2a)
Comparing our equation with the standard quadratic form (ax^2 + bx + c = 0), we have:
a = -16, b = 184, and c = 0.
Substituting these values into the quadratic formula, we get:
t = (-184 ± √(184^2 - 4*(-16)*0)) / (2*(-16))
t = (-184 ± √(33856)) / (-32)
Finally, solving for t gives us two possible solutions:
t = (-184 + √(33856)) / (-32)
t = (-184 - √(33856)) / (-32)
Calculating these values will give us the times for the projectile to return to Earth.