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:

**Use the Newton's Second Law equation Σ***F*=to find a function for the acceleration**ma****Integrate**the**acceleration function**to get velocity (don't forget about the integration constant!)**Integrate the velocity**to get the position (if required)**Calculate**the position and/or velocity at a**specific time**of interest (if required)

A diagram like this one can help you remember the relationships between the variables:

$\mathit{P}\begin{array}{c}{\mathbf{\leftarrow}}\\ {\mathbf{\to}}\end{array}\underset{\frac{\mathit{d}}{\mathit{d}\mathit{t}}}{\overset{{\mathbf{\int}}{\mathit{d}}{\mathit{t}}}{\mathit{V}}}\begin{array}{c}{\mathbf{\leftarrow}}\\ {\mathbf{\to}}\end{array}\underset{\mathit{F}\mathbf{=}\mathit{m}\mathit{a}}{\mathit{A}\mathbf{,}\mathit{F}}$

Remember the **power rule** of integration.

To integrate,

$\overline{){\mathbf{\int}}{{\mathbf{t}}}^{{\mathbf{n}}}{\mathbf{}}{\mathbf{d}}{\mathbf{t}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{n}\mathbf{+}\mathbf{1}}{{\mathbf{t}}}^{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{+}}{\mathbf{C}}}$, where **C** is the constant of integration.

The **velocity** function is the **integral** of the acceleration function,** a(t)**. We're not given

The object's **position **function is the expression for **x****( t)**, which is the integral of the velocity function

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 F_{x} = F_{0} (1 − t/T), which decreases from F_{0} 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.