57 Human Locomotion on Earth

colliding with Earth

In the previous chapter we learned that the body can achieve a change in by initiating a collision with another object and that we can analyze the change in velocity for each object using . In our examples so far the other object had similar mass to the body and experienced a similar change in velocity, but in most everyday situations the body achieves locomotion by interacting with the Earth (or a floor or building firmly attached to the Earth). In those cases the change in velocity for the much larger object is typically negligible.

Everyday Example: Locomotion by Collision with Earth

A person of about 65 kg mass starts from rest and pushes off the ground to take a step and reach a of 1.2 m/s at the end of the step. What is the resulting velocity of the Earth?

We will start with the applied to just two objects.

    \begin{equation*} m_{1,i}\bold{v_{1,i}} +m_{2,i}\bold{v_{2,i}} = m_{1,f}\bold{v_{1,f}} +m_{2,f}\bold{v_{2,f}} \end{equation*}

The initial velocity of both objects is zero:

    \begin{equation*} 0 + 0 = m_{1,f}\bold{v_{1,f}} + m_{2,f}\bold{v_{2,f}} \end{equation*}

Now we subtract the initial momentum of the Earth (object #2) to the left side:

    \begin{equation*} - m_{2,f}\bold{v_{2,f}} = m_{1,f}\bold{v_{1,f}} \end{equation*}

Divide both sides by the mass of the earth and multiply both sides by (-1) to isolate the final velocity of the earth:

    \begin{equation*} \bold{v_{2,f}}= \frac{-m_{1,f}\bold{v_{1,f}}}{m_{2,f}} \end{equation*}

The mass of the earth is 5.6 \times 10^{24}\, \bold{kg} so we have:

    \begin{equation*} \bold{v_{2,f}}= \frac{(-65\,\bold{kg})(1.2\,\bold{m/s})}{5.6 \times 10^{24}\, \bold{kg}} = - 1.4\times 10^{-23} \, \bold{m/s} \end{equation*}

In units of m/s that’s a final Earth velocity of 14 after 22 zeros following the decimal or about the width of one atom per million years. The answer is negative because the Earth moves in the opposite direction as the person and we entered a positive value for their .

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Body Physics 2.0 by Lawrence Davis is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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