How to find maximum height of a body

Question

The height of a body moving vertically is given by s=1/2gt²+v₀t+s₀ for g>0 with s in meters and t in seconds. What is the body's maximum height?

This is an expression, not necessarily numerical. How do I go about finding this? 

Answer 1

I believe your formula should have a negative on t² term:

s = (-1/2)gt² + v₀t + s₀

 

To find any extreme value, you'll take the first derivative, set it equal to zero and solve. g, v0, and s0 are constants, so treat them like numbers, not variables.

 

s' = (-1/2)g(2t) + v₀

s' = -gt + v₀ = 0

The critical value is t = v₀/g

 

You can then plug this into the original equation to solve for the maximum height.

 

s_max = (-1/2)g(v₀/g)² + v₀(v₀/g) + s0

s_max = - v₀/(2g) + v₀²/g + s₀

 

You can combine by using a common denominator of 2g if you like.

Answer 2

The maximum height should be: (2g*So-Vo^2)/2g