Example 4

Newton's law, F = ma, states that force equals mass times acceleration. Suppose a constant force F is applied along the y-axis to an object of constant mass m. Then the position y of the object is governed by the second order differential equation

The general solution of this equation is found by integrating twice,

Setting t = 0 we see that the constants u₀ and y₀ are just the velocity and position at time f = 0. Thus the motion of the object is known if we know its initial position y₀ and velocity y₀.

If the force F(t) varies with time we have the differential equation

The general solution can still be found by integrating twice, and the motion will still be determined by the initial position and velocity. Suppose for example that F(t) = t², and y₀ = 5, v₀ = 1 at time f = 0. Then