# MAT275 Modeling with First Order

A body of mass 6 kg is projected vertically upward with an initial velocity 25 meters per second.

We assume that the forces acting on the body are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude c|v(t)| where c=0.35kgs and v(t) is the velocity of the ball at time t. The gravitational constant is g=9.8m/s2.

a) Find a differential equation for the velocity v:
dvdt=−9.8−0.356|v|

b) Solve the differential equation in part a) and find a formula for the velocity at any time t:
v(t)= 168.9−194.1e−0.058t

Find a formula for the position function at any time t, if the initial position is s(0)=0:
s(t)= 3346.5e−0.058t−168.9t+3346.5

How does this compare with the solution to the equation for velocity when there is no air resistance?

If c=0, then v(t)=25−9.8t, and if s(0)=0, then s(t)=25t−4.9t2.
We then have that v(t)=0 when t≈2.551, and s(2.551)≈31.888,
and that the positive t solution to s(t)=0 is t≈5.102, which leads to v(5.102)=−25 meters per Requirements: Medium | .doc file

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