This problem requires us to find the **position**** **of the object after a force giving it constant acceleration is applied.

This is a **Forces with Calculus** type of problem since we have the force as a function of time. We'll 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:

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

Since we're looking for the object's **position**, we want to find an expression for **x****( t)**, which is the integral of the velocity function

The **velocity** function is the **integral** of the acceleration function,** a(t)**. We're not given that, either, but we are given a function for

A mysterious rocket-propelled object of mass 45.5 kg is initially at rest in the middle of the horizontal, frictionless surface of an ice-covered lake. Then a force directed east and with magnitude *F*(*t*) = (16.1 N/s)*t* is applied.

How far does the object travel in the first 5.25 s after the force is applied?

Frequently Asked Questions

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. If you need more Forces with Calculus practice, you can also practice Forces with Calculus practice problems.