This problem relates to projectiles without air resistance, but using DEs Solve the IVP to find an expression for v(t) in terms of g and t(no m dependence)
1st: Given mv'(t)=-mg ; v(0)=v0
2nd: v'(t)=-g (dividing by m, since there is no mass dependence)
3rd: integrate v(t)=-9.8t
I'm stuck here, because I am to use the result from v(t) and solve for the IVP: x'(t)=v(t) ; x(0)=0 (height above ground of the projectile) in terms of v0,g, and t.
A projectile is shut up vertically without taking air resistance into account, considering a basic vertical throw.After being shot, the only force acting upon the projectile is gravity -mg, with negative sign indicating gravity act downwards. Thus Newton's Law gives ma(t)=-mg. With a(t)=v'(t) and use v(0)=vo(not) as initial condition this gives the first order initial value problem for velocity:
mv'(t)=-mg, v(0)=vo....my original post goes from here.....
First integrate both sides to get .
Applying the initial condition , we have .
Thus, .
Since , it follows that if we integrate both sides of the equation, we have
.
Now, supposing that , we have .
Thus, .
In your first post, I saw that you let . So it turns out that the equation you're interested in is .
(If you're familiar with physics, we take note that this is the kinematics equation , where we have and )