Relative Velocity

Airplane Velocity Vectors

Relative Motion... The plane is moving north in the frame of reference attached to the air: Vp, a is the velocity of the plane w.r.t. the air. Air Vp,a

Relative Motion... But suppose the air is moving east in the IRF attached to the ground. Va,g is the velocity of the air w.r.t. the ground (i.e. wind). Air Vp,a Va,g

Relative Motion... What is the velocity of the plane in a frame of reference attached to the ground? Vp,g is the velocity of the plane w.r.t. the ground. Vp,g

Relative Motion... Vp,g = Vp,a + Va,g is a vector equation relating the airplane’s velocity in different reference frames. Va,g Vp,a Vp,g

Airplane ACT The velocity of an airplane relative to the air is 100 km/h, due north. A crosswind blows from the west at 20 km/h. What is the velocity of the plane relative to the ground? Vp,g Va,g Vp,a 102 km/h, 79o

Boat in River Velocity

Motorboat ACT Consider a motorboat that normally travels 10 km/h in still water. If the boat heads directly across the river, which also flows at a rate of 10 km/h, what will be its velocity relative to the shore? When the boat heads cross-stream (at right angles to the river flow) its velocity is 14.1 km/h, 45 degrees downstream .

Preflight Responses Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across a river in the least time. Relative to the water, Beth (B) swims perpendicular to the flow of the river (shown by the horizontal arrow in the figure), Ann (A) swims upstream, and Carly (C) swims downstream. Which swimmer wins the race? 11% 26% 63%

Boat Velocity (1) Which boat takes the shortest path to the opposite shore? (2) Which boat reaches the opposite shore first? (3) Which boat provides the fastest ride?

Perpendicular Velocities ACT Vx = 2 m/s 1 m Vy = 0.5 m/s How long does it take the ladybug to crawl to the opposite side of the paper? This is independent of vx!!!!!!!!!!

Independence of Velocities If a boat heads perpendicular to the current at 20 m/s relative to the river, how long will it take the boat to reach the opposite shore 100 m away in each of the following cases? Current speed = 1 m/s Current speed = 5 m/s Current speed = 10 m/s Current speed = 20 m/s

Swimmer ACT You are swimming across a 50m wide river in which the current moves at 1 m/s with respect to the shore. Your swimming speed is 2 m/s with respect to the water. You swim across in such a way that your path is a straight perpendicular line across the river. How many seconds does it take you to get across ? (a) (b) (c) 1 m/s 50 m 2 m/s

solution y Choose x axis along riverbank and y axis across river x The time taken to swim straight across is (distance across) / (vy ) Since you swim straight across, you must be tilted in the water so that your x component of velocity with respect to the water exactly cancels the velocity of the water in the x direction: 1 m/s 1m/s y 2 m/s m/s x

solution So the y component of your velocity with respect to the water is So the time to get across is m/s m/s 50 m y x

Frame of Reference