Differences
This shows you the differences between two versions of the page.
Next revision | Previous revision | ||
183_notes:examples:relativemotion [2014/07/11 02:29] – created caballero | 183_notes:examples:relativemotion [2014/11/16 08:05] (current) – pwirving | ||
---|---|---|---|
Line 10: | Line 10: | ||
* The pilot intends to fly due west. | * The pilot intends to fly due west. | ||
- | * The plane experiences a crosswind with a speed of 10.0 $\dfrac{m}{s}$, | + | * The plane experiences a crosswind with a speed of $|v_{a/g}| = 10.0 \dfrac{m}{s}$, |
=== 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 === | ||
Line 26: | Line 26: | ||
* 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/ | ||
- | |||
==== Solution ==== | ==== Solution ==== | ||
+ | |||
+ | The problem can be described vectorially using the relative velocity equation: | ||
+ | |||
+ | $$\vec{v}_{p/ | ||
+ | |||
+ | The pilot requires that the velocity of the plane with respect to the ground be directed due west. If you measure positive $\theta$ counterclockwise with respect to the east, the plane' | ||
+ | |||
+ | $$\langle -|v_{p/ | ||
+ | |||
+ | We can break this vector equation into two scalar equations: | ||
+ | |||
+ | $$-|v_{p/ | ||
+ | $$0=|v_{p/ | ||
+ | |||
+ | where ${v}_{p/ | ||
+ | |||
+ | $${v}_{p/ | ||
+ | |||
+ | You can rewrite the above equation by using what the unit vector components are equal to (in the previous two equations), | ||
+ | |||
+ | |||
+ | $$\left(-\dfrac{|v_{p/ | ||
+ | $${|v_{p/ | ||
+ | |||
+ | So, the speed that the plane has with respect to the ground is slower than its air speed, which agrees with the representation above. | ||
+ | $${|v_{p/ | ||
+ | |||
+ | 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, | ||
+ | |||
+ | $$\tan \theta = \dfrac{|v_{a/ | ||
+ | |||
+ | Hence, | ||
+ | |||
+ | $$\theta = \tan^{-1} \left(\dfrac{|v_{a/ | ||
+ | |||
+ | which is 2.5$^{\circ}$ north of west or 177.5$^{\circ}$ from east measured counterclockwise. | ||
+ | |||
+ |