Forces with Calculus Video Lessons

Concept

Problem: 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?

FREE Expert Solution

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:

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 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:

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

Remember the power rule of integration:

, 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 v(t). The only information we have about the object's velocity is that it starts from rest, so we need another step, though!

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 force, F(t). So here's our process:

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Problem Details

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?