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.

well, just plug in your numbers and solve for t when the height is zero:

-16t^2 + 184t = 0

just a good old quadratic equation.

12.9

To find the time for a projectile to return to Earth, we need to find the value of t when the height of the projectile, s, is equal to zero.

Given that the height of the projectile is given by the formula s = v0t - 16t^2, we can set this equation equal to zero and solve for t.

So, we have:
0 = v0t - 16t^2

Now, substitute the given value of the initial velocity, v0 = 184 ft/s, into the equation to get:
0 = 184t - 16t^2

Next, rearrange the equation to the form of a quadratic equation:
16t^2 - 184t = 0

Now, factor out a common term of 8t:
8t(2t - 23) = 0

To solve for t, we set each factor equal to zero:
8t = 0 or 2t - 23 = 0

From the first equation, we get:
t = 0

However, this value of t does not make practical sense in the context of the problem since it represents the time when the projectile was fired, not when it returns to Earth.

So, we solve the second equation to find the valid value of t:
2t - 23 = 0

Adding 23 to both sides:
2t = 23

Dividing both sides by 2:
t = 23/2

The time for the projectile to return to Earth is t = 23/2, which is equal to 11.5 seconds.