calculus question - fundamental theorem

Question

An object is launched upward from a platform. Its height is a function of time, h(t). The rate of change of velocity is d2h/dt2= g,

where g is an unknown constant.

The velocity at time t = 0 is v0. Find a formula for velocity using g, v0 and t.

dh/dt = _____________

The height at time zero is h0.

Find a formula for height using g, v0, h0 and t.

h(t) =______________

Answer 1

Since dh/dt is velocity, we can use the initial conditions for t = 0, v = v₀ to get C = v₀

meaning the answer to the first question is dh/dt = gt + v₀ then:

h = 1/2gt² + v₀t + C

Using the initial conditions that at t = 0, h = h₀ then C = h₀ therefore:

h(t) = 1/2gt² + v₀t + h₀