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.

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What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Forces with Calculus concept. You can view video lessons to learn Forces with Calculus. Or if you need more Forces with Calculus practice, you can also practice Forces with Calculus practice problems.