Forces with Calculus Video Lessons

Concept

Problem: At t = 0, an object of mass m is at rest at x = 0 on a horizontal, frictionless surface. Starting at t = 0, a horizontal force Fx = F0e−t/T is exerted on the object.(a) Find an expression for the object's velocity at an arbitrary later time t.(b) What is the object's velocity after a very long time has elapsed?￼

FREE Expert Solution

This problem requires us to find the velocity function and evaluate its magnitude after a very long time given a force function.

This is a Force with Calculus type of problem since we have the force as a function of time. 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. Calculate the velocity at a specific time of interest.

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

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

In step 2, we need to recall how to integrate an exponential.

, 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

At t = 0, an object of mass m is at rest at x = 0 on a horizontal, frictionless surface. Starting at t = 0, a horizontal force Fx = F0e−t/T is exerted on the object.
(a) Find an expression for the object's velocity at an arbitrary later time t.
(b) What is the object's velocity after a very long time has elapsed?￼