This problem requires us to find the velocity and position expressions for an object at a specific time given a force function.
For a Force with Calculus type of problem, we follow these steps:
A diagram like this one can help you remember the relationships between the variables:
Remember the power rule of integration.
, where C is the constant of integration.
The velocity function is the integral of the acceleration function, a(t). We're not given a(t), but we are given a function for force, F(t).
The object's position function is the expression for x(t), which is the integral of the velocity function v(t).
So here's our process:
At t = 0, an object of mass m is at rest at x = 0 on a horizontal, frictionless surface. A horizontal force Fx = F0 (1 − t/T), which decreases from F0 at t = 0 to zero at t = T, is exerted on the object.
(a) Find an expression for the object's velocity at time T.
(b) Find an expression for the object's position at time T.