Relative velocity Velocity always defined relative to reference frame. All velocities are relative Relative velocities are calculated by vector addition/subtraction.

Example 3.8 1.067 hours = 1 hr. and 4 minutes 187.4 mph A plane that is capable of traveling 200 m.p.h. flies 100 miles into a 50 m.p.h. wind, then flies back with a 50 m.p.h. tail wind. How long does the trip take? What is the average speed of the plane for the trip? 1.067 hours = 1 hr. and 4 minutes 187.4 mph

Relative velocity in 2-d Sum velocities as vectors velocity relative to ground = velocity relative to medium + velocity of medium. vbe = vbr + vre river wrt earth Boat wrt earth boat wrt river

pointed perpendicular to stream travels perpendicular to stream 2 Cases pointed perpendicular to stream travels perpendicular to stream

Example Problem A plane flies due north with an airspeed of 50 m/s, while the wind is blowing 15 m/s due East. What is the speed and direction of the plane with respect to the earth? What do we know? “Airspeed” means the speed of the plane with respect to the air. “wind blowing” refers to speed of the air with respect to the earth. What are we looking for? “speed” of the plane with respect to the earth. We know that the speed and heading of the plane will be affected by both it’s airspeed and the wind velocity, so… just add the vectors.

Example Problem (cont.) So, we are adding these vectors…what does it look like? Draw a diagram,of the vectors tip to tail! Solve it! This one is fairly simple to solve once it is set up…but, that can be the tricky part. Let’s look at how the vector equation is put together and how it leads us to this drawing. N θ

How to write the vector addition formula middle same first last Note: We can use the subscripts to properly line up the equation. We can then rearrange that equation to solve for any of the vectors. Always draw the vector diagram, then you can solve for any of the vector quantities that might be missing using components or even the law of sines.

The engine of a boat drives it across a river that is 1800m wide. Crossing a River The engine of a boat drives it across a river that is 1800m wide. The velocity of the boat relative to the water is 4.0m/s directed perpendicular to the current. The velocity of the water relative to the shore is 2.0m/s. (a) What is the velocity of the boat relative to the shore? (b) How long does it take for the boat to cross the river? (c) How far downstream does the boat come to ground? 8

What do these subscripts means? BS = Boat relative to Shore BW = Boat relative to Water WS = Water relative to Shore 9

θ = Cos-1 ( X / H) Cos-1 (2 / 4.5) = 63o 10

How far downstream does the boat come to ground? Dx=1800 /Tan (63) = θ = Cos-1 ( X / H) Cos-1 (2 / 4.5) = 63o How far downstream does the boat come to ground? Dx=1800 /Tan (63) = = 900m. Also, 450s x 2 m/s = 900m 11

Relative Motion (why is it important) Another example of relative motion is the motion of airplanes. Runways are fixed in the reference frame of the earth, while airplanes fly in a reference attached to the air. On landing the airplane needs to transition from the motion in the air to motion on the ground. This can be tricky when there are strong cross winds with respect to the runway.

What is the resulting ground speed? Example 3.9 An airplane is capable of moving 200 mph in still air. The plane points directly east, but a 50 mph wind from the north distorts his course. What is the resulting ground speed? What direction does the plane fly relative to the ground? 206.2 mph 14.0 deg. south of east

What is the plane’s resulting ground speed? Example 3.10 An airplane is capable of moving 200 mph in still air. A wind blows directly from the North at 50 mph. The airplane accounts for the wind (by pointing the plane somewhat into the wind) and flies directly east relative to the ground. What is the plane’s resulting ground speed? In what direction is the nose of the plane pointed? 193.6 mph 14.5 deg. north of east