A 30 kg rail cart with a wind sail is on a frictionless track that runs northeast and southwest. A wind applies a force vector of F = [100 50] to the cart. Calculate the acceleration of the cart.

Question

A 30 kg rail cart with a wind sail is on a frictionless track that runs northeast and southwest. A wind applies a force vector of [latex]\vec{F}=\left [ \begin{matrix} 100 \ 50 \end{matrix} ight ] [/latex] to the cart. Calculate the acceleration of the cart.

Accepted Answer

Since the rail cart is on a track, only the amount of force in the direction of the track will create acceleration in the direction of the track. The track has direction vector [latex]\vec{d}=\left [ \begin{matrix} 1 \ 1 \end{matrix} ight ] [/latex]. Thus, the force vector in that direction is

[latex]proj_{\vec{d}}(\vec{F})=\frac{\vec{d}.\vec{F}}{\left\|\vec{d} ight\|^{2} }\vec{d} =\frac{150}{2}\left [ \begin{matrix} 1 \ 1 \end{matrix} ight ] =\left [ \begin{matrix} 75 \ 75 \end{matrix} ight ] [/latex]

Thus, the amount of force in the direction of the track is

[latex] \left\|proj_{\vec{d}}(\vec{F}) ight\| =\sqrt{(75)^{2}+(75)^{2}} \approx 106N[/latex]

Consequently, the acceleration of the cart along the track will be [latex]a=\frac{F}{m} \approx 3.53m/s^{2}[/latex].

Question 1.5.7: A 30 kg rail cart with a wind sail is on a frictionless trac...

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