Forces with Calculus Video Lessons

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# Problem: A particle of mass m moving along the x-axis experiences the net force Fx = ct, where c is a constant. The particle has velocity v0x at t = 0.Find an algebraic expression for the particle's velocity vx at a later time t.

###### FREE Expert Solution

This problem requires us to find an expression for velocity as a function of time, given an expression for the force on an object.

In this problem, we'll follow these steps:

1. Use the Newton's Second Law equation ΣF=ma to find a function for the acceleration
2. Integrate the acceleration function to get velocity (don't forget about the integration constant!)
3. Integrate the velocity to get the position (if required)
4. Calculate the position at a specific time of interest (if required)

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

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

Remember the power rules of integration.

To integrate,

, 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). So here's our process:

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###### Problem Details

A particle of mass m moving along the x-axis experiences the net force Fx = ct, where c is a constant. The particle has velocity v0x at t = 0.

Find an algebraic expression for the particle's velocity vx at a later time t.