 Suppose you are a passenger in a car on a perfectly level and straight road, moving at a constant velocity.  Your velocity relative to the pavement might be 60 mph. Topic 2.1 Extended G – Relative velocity  Your velocity relative to the driver of your car is zero.  Your velocity relative to an oncoming car might be 120 mph.  Your velocity can be measured relative to any reference frame.  Which reference frame do we usually measure relative to? A

Topic 2.1 Extended G – Relative velocity  Consider two cars, A and B, shown below. A B  Suppose both cars are moving at the constant velocities v A = + 20 x mph and v B = + 20 x mph as shown. v AB = 0 ^ ^  Suppose you are in car A.  As far as you are concerned, your velocity relative to car B is 0, and we write, which means "the velocity of A relative to B."  Of course, v BA = 0 too. FYI: Beware of the subscripting. In general, v AB is different from v BA.  If we define v AB = v A - v B, note that we get the right result of 0, since v AB = v A - v B = + 20 x mph ^ x mph ^ = 0 x mph. ^ FYI: At this point, it may not be clear that our "test formula" is correct, since v AB = v B - v A would also work.

Topic 2.1 Extended G – Relative velocity  Consider two cars, A and B, shown below. A B  Suppose car A is moving at v A = + 20 x mph and car B is moving at v B = + 40 x mph as shown. ^ ^  Suppose you are in car A.  As far as you are concerned, your velocity relative to car B is -20 x mph, because you seem to be moving backwards relative to B. ^ v AB = v A - v B = + 20 x mph ^ x mph ^ = - 20 x mph. ^  In this case FYI: This should almost convince you that the formula works.  If you are in B, your velocity relative to A is v BA = v B - v A = + 40 x mph ^ x mph ^ = + 20 x mph. ^

Topic 2.1 Extended G – Relative velocity  Consider two cars, A and B, shown below.  Suppose car A is moving at v A = + 20 x mph and car B is moving at v B = - 40 x mph as shown. ^ ^  Suppose you are in car A.  As far as you are concerned, your velocity relative to car B is +60 x mph, because you seem to be moving forward relative to B. ^ v AB = v A - v B = + 20 x mph ^ x mph ^ = + 60 x mph. ^  In this case FYI: Are you convinced that the formula works? I hope so.  If you are in B, your velocity relative to A is v BA = v B - v A = - 40 x mph ^ x mph ^ = - 60 x mph. ^ A B v AB = v A - v B velocity of A relative to B v BA = v B - v A velocity of B relative to A

Topic 2.1 Extended G – Relative velocity  Now that we are convinced that our formula works in 1D, we can generalize it to 2D. v AB = v A - v B velocity of A relative to B x y A B  Suppose car B has velocity v B = - 20 y (mph). = + 40 x y (mph) |v AB | 2 = |v AB | = mph (relative approach speed) ^ ^  Suppose you are in car A, and v A = 40 x (mph).  Then v AB = v A - v B v AB = 40 x + 20 y (mph)  So how fast are the two cars actually approaching?

Topic 2.1 Extended G – Relative velocity  Besides having two particles (like cars) moving relative to one another, you can have one particle (like a boat) and a fluid (like a river) moving relative to the container (the ground).  Suppose a boat is capable of moving at v BW = 2 knots in still water, as shown below.  Now suppose that the still water is actually a river moving to the right at v WG = 1 knot: v WG = 1 kn  Its speed relative to the ground will be v BG = ±2 knots.  The boat's speed relative to the ground will now be v BG = + 3 knots or v BG = - 1 knot, depending on direction. v BG = + 3 kn v BG = - 1 kn v BW = - 2 kn v BW = + 2 kn  The relationship is clearly v BW + v WG = v BG. v BW + v WG = v BG the boat equation

Topic 2.1 Extended G – Relative velocity  The boat equation also generalizes easily to 2D. i j v WG Destination v BW desired line  Suppose your boat can do v BW = 2 kn.  Suppose the river flows at v WG = 1 kn. v BW 2 kn v WG 1 kn v BG 2.24 kn  Suppose you want to reach the direct opposite shore.  If you aim directly to the opposite shore the current will carry you downstream. θ  A velocity diagram looks like this: |v BG | 2 = θ |v BG | = 2.24 kn v BW + v WG = v BG boat equation Origin  The magnitude and direction of v BG is = tan -1 (1/2) = 26.6° actual line

Topic 2.1 Extended G – Relative velocity  In order to get to the destination, you must steer upstream at a sufficient angle to oppose the current. i j v water Destination v boat desired line  What angle will this be?  At first thought, you might think that it would be 26.6°. v BW 2 kn v WG 1 kn v BG 1.73 kn  Work from basic principles, and never assume.  If you aim 30° upstream of your destination, you will just counter the current. θ  A velocity diagram looks like this: |v BG | = 2 2 θ |v BG | = 1.73 kn θ = sin -1 (1/2) = 30° FYI: The boat is said to be "crabbing" because from above it appears to be moving sideways toward its destination. An airplane does the same thing. The fluid in this case is AIR. The plane has a similar equation to the boat: v PA + v AG = v PG plane equation