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183_notes:examples:relativemotion [2014/07/11 02:37] – [Solution] caballero | 183_notes:examples:relativemotion [2014/11/16 08:05] (current) – pwirving | ||
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=== Lacking === | === Lacking === | ||
- | * The top speed of a Boeing 747 is unknown, but can be [[http:// | + | * The top speed of a Boeing 747 is unknown, but can be [[http:// |
=== Approximations & Assumptions === | === Approximations & Assumptions === | ||
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* The velocities of the plane relative to the air, the air relative to the ground, and the plane relative to the ground are represented in the following diagram. | * The velocities of the plane relative to the air, the air relative to the ground, and the plane relative to the ground are represented in the following diagram. | ||
- | <WRAP todo>Add vector addition diagram</ | + | {{ 183_notes: |
* The relative velocity equation for three objects is: $\vec{v}_{A/ | * The relative velocity equation for three objects is: $\vec{v}_{A/ | ||
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==== Solution ==== | ==== Solution ==== | ||
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$$\langle -|v_{p/ | $$\langle -|v_{p/ | ||
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+ | We can break this vector equation into two scalar equations: | ||
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+ | $$-|v_{p/ | ||
+ | $$0=|v_{p/ | ||
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+ | where ${v}_{p/ | ||
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+ | $${v}_{p/ | ||
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+ | You can rewrite the above equation by using what the unit vector components are equal to (in the previous two equations), | ||
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+ | $$\left(-\dfrac{|v_{p/ | ||
+ | $${|v_{p/ | ||
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+ | So, the speed that the plane has with respect to the ground is slower than its air speed, which agrees with the representation above. | ||
+ | $${|v_{p/ | ||
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+ | The angle the compass should read can be determined from the above representation. The tangent of the angle (as measured from the negative $x$-axis is given by, | ||
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+ | $$\tan \theta = \dfrac{|v_{a/ | ||
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+ | Hence, | ||
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+ | $$\theta = \tan^{-1} \left(\dfrac{|v_{a/ | ||
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+ | which is 2.5$^{\circ}$ north of west or 177.5$^{\circ}$ from east measured counterclockwise. | ||
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