Relative motion
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Relative Motion
Imagine two ships travelling. Ship A is travelling at 10 kmh-1 due North and ship B is travelling at 5 kmh-1 due East. Suppose you are standing on the deck of ship B and look at ship A. You will not see ship A travelling a 10 kmh-1 due North, because you are yourself travelling due North. What you see is the velocity of A relative to B. We use the symbol for this. [Diagram goes here - download the original to see it.] The example illustrates that we would expect the velocity of A relative to B (AVB) to be the vector joining the tip of B to A. Using the addition law for vectors VB + AVB = VA \ AVB = VA - VB In general objects will be traveling in a certain direction (given by a bearing) and at a certain speed. [Diagram goes here - download the original to see it.] Thus, the relative velocity of A to B is the vector joining the tip of VB to the tip of VA Further: AVB = VA - VB The meaning of AVB is that A appears to B to be moving with a velocity in this direction and of this magnitude - that is, this is what B sees by assuming that he is stationary. When solving problems involving relative velocities: 1. Convert all velocities to component form - i.e. using i , j notation or equivalent. 2. Use physical intuition to draw a correct diagram. Remember AVB is the vector joining B to A, so a diagram like this: [Diagram goes here - download the original to see it. Does not give a relative velocity. This diagram shows the resultant of adding VA to V B. 1. Problems involving velocities are not direct problems involving distances. A diagram showing relative velocities does not show positions and distances and if these are required a separate diagram should be drawn. Velocities are general vectors not position vectors. This means that in a diagram they do not have to be anchored anywhere. They can be shifted about. If they retain the same direction and magnitude then they are the same vector. [Diagram goes here - download the original to see it.] These are all the same vector - there is no origin in this diagram. Example 1. A ship is travelling at 10 kmh-1 due North and a dove is travelling at 5 kmh-1 due East. What is the apparent speed & direction of the dove to an observer on the ship? Solution [Diagram goes here - download the original to see it.] VS = 10j VD = 5 i DVS = VD - VS = 10j - 5i Apparent speed = [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Bearing z = 90 + q = 90 + 26.6 = 116.6° Example 2 A female skater is travelling at 3 ms-1 on a bearing of 70° and her partner is travelling at 4 ms-1 on a bearing of 140°. What is the velocity of the female skater relative to the male skater? Solution [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] F V M = VF - VM = (3 sin 70 i + 3 cos 70 j ) - (v sin 40 i - 4 cos 40 j ) = (3 sin 70 - 4 sin 40 ) i + (3 cos 70 + 4 cos 40) j = 0.2479 i - 4.0902 j [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] In the following example, you are given the relative velocity, and one other velocity and asked to find the other velocity. Use the equation AVB = VA - VB , substitute in, and solve. Example 3. A toy plane flies at 11 kmh-1 on a bearing of 30°. The wind appears to be coming from 80° at 20 kmh-1. What is the real velocity of the wind? [Diagram goes here - download the original to see it.] The apparent velocity of the wind is the velocity of the wind relative to the toy plane. W VP = VW - VP \-20 sin 80 i - 20 cos 80 j = VW - (11 sin 30 i + 11 cos 30 j ) VW = (-20 sin 80 + 11 sin 30 ) i - (20 cos 80 - 11 cos 30 ) j = -14.196 i + 6.053 j speed = [Equation goes here - download the original to see it.] = 15.4 kmh-1 [Equation goes here - download the original to see it.] bearing = 360 - 66.9 = 293.1°
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Contents of
Relative motion
1 Relative Motion
2 Interception
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