Height of a Projectile.

The height of a projectile is described by the equation:

s(t)=144t-16t^(2)

Where t is in seconds and s(t) is the height of the projectile at time t. Find when the projectile hits the ground.

projectile hits the ground when s(t) = 0 meaning the height off the ground is zero.

0 = 144t -16 t^2

Factor out the t:
0 = t(144 -16t)

set each part = to zero

t = 0 the projectile wasn't launched yet

and 144- 16t = 0
solve for t to find out how long the projectile was in the air before hitting the ground.

To find when the projectile hits the ground, we need to determine the time at which the height, s(t), is equal to 0.

Given the equation for the height of the projectile: s(t) = 144t - 16t^2

Set s(t) equal to 0:

0 = 144t - 16t^2

Rearrange the equation to make it easier to solve for t:

16t^2 - 144t = 0

Divide both sides of the equation by 16 to simplify:

t^2 - 9t = 0

Factor out "t" from the equation:

t(t - 9) = 0

Now, we have two possible solutions for t:

t = 0 or t - 9 = 0

Solving the second equation, we find:

t - 9 = 0
t = 9

Therefore, the projectile hits the ground at t = 9 seconds.

To find when the projectile hits the ground, we need to determine the time t at which the height of the projectile, described by the equation s(t), is equal to zero.

Given the equation for the height of the projectile as:

s(t) = 144t - 16t^2

We set s(t) to zero and solve for t:

0 = 144t - 16t^2

To make it easier to solve, we can first divide the equation by 16:

0 = 9t - t^2

Rearranging the equation, we get:

t^2 - 9t = 0

Now, factoring out t:

t(t - 9) = 0

From this equation, we can see that either t = 0 or (t - 9) = 0.

If t = 0, it means the projectile started at time t = 0. This is not the time at which it hits the ground, so we can ignore this solution.

The other solution is t - 9 = 0, which means t = 9. Therefore, the projectile hits the ground after 9 seconds.

Hence, the projectile hits the ground at t = 9 seconds.