1. Preview Section 1 Introduction to Vectors Section 2 Vector Operations
Section 3 Projectile Motion
Section 4 Relative Motion

2. Relative Motion Click below to watch the Visual Concept.

3. Frames of Reference
A falling object is shown from two different frames of reference:
the pilot (top row)
an observer on the ground (bottom row)
You might want to discuss the old movies of WW I and II pilots after they released the bombs. They always turned to the right or left or sharply upward. This was particularly important if they were flying low. Students should be able to see why in the lower sequence of pictures.

4. Relative Velocity vac = vab + vbc
vac means the velocity of object “a” with respect to frame of reference “c”
Note: vac = -vca
When solving relative velocity problems, follow this technique for writing subscripts.

5. Sample Problem
A boat is traveling downstream. The speed of the boat with respect to Earth (vbe) is 20 km/h. The speed of the river with respect to Earth (vre) is 5 km/h. What is the speed of the boat with respect to the river?
Solution:
vbr = vbe+ ver =
vbe + (-vre) = 20 km/h + (-5 km/h)
vbr = 15 km/h
This can more easily be solved using a diagram or common sense, but it is useful to show the students how to manipulate the equation to switch velocities from one frame of reference to another.

6. Classroom Practice Problem
A passenger at the rear of a train traveling at 15m/s relative to the earth throws a baseball with a speed of 15m/s in the direction opposite the motion of the train. What is the velocity of the baseball relative to Earth as it leave the thrower’s hand?
Answer: 0 m/s
You might want to work through Problem F in the text before having students work on this problem.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
For this problem, vpg = vpa + vag Since both of these values are provided, it is a simple vector addition with two right-angle vectors. It might be a good idea to use the component method to add these, since it is unusual to have a right angle between the two vectors. It will also reinforce the notion of breaking vectors down into components before recombining them to get a resultant.

7. Classroom Practice Problem
A spy runs from the front to the back of an aircraft carrier at a speed of 3.5m/s. If the aircraft carrier is moving forward at 18,0 m/s, how fast does the spy appear to be running when viewed by an observer on a nearby stationary submarine?
Answer: 14.5 m/s in the direction that the aircraft carrier is moving
You might want to work through Problem F in the text before having students work on this problem.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
For this problem, vpg = vpa + vag Since both of these values are provided, it is a simple vector addition with two right-angle vectors. It might be a good idea to use the component method to add these, since it is unusual to have a right angle between the two vectors. It will also reinforce the notion of breaking vectors down into components before recombining them to get a resultant.

8. Classroom Practice Problem
A ferry is crossing a river. If the ferry is headed due north with a speed of 2.5m/s relative to the water and the river’s velocity is 3.0m/s to the east, what will the boat’s velocity be relative to the Earth? (Remember to include the direction in describing the velocity
Answer: 3.9 m/s at 40.0° north of east
You might want to work through Problem F in the text before having students work on this problem.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
For this problem, vpg = vpa + vag Since both of these values are provided, it is a simple vector addition with two right-angle vectors. It might be a good idea to use the component method to add these, since it is unusual to have a right angle between the two vectors. It will also reinforce the notion of breaking vectors down into components before recombining them to get a resultant.

9. Classroom Practice Problem
A passenger at the rear of a train traveling at 15m/s relative to the earth throws a baseball with a peed of 15m/s in the direction opposite the motion of the train. What is the velocity of the baseball relative to Earth as it leave the thrower’s hand?
Answer: km/h at 40.1° north of east
You might want to work through Problem F in the text before having students work on this problem.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
For this problem, vpg = vpa + vag Since both of these values are provided, it is a simple vector addition with two right-angle vectors. It might be a good idea to use the component method to add these, since it is unusual to have a right angle between the two vectors. It will also reinforce the notion of breaking vectors down into components before recombining them to get a resultant.

10. Classroom Practice Problem
A pet store supply truck moves at 25.0m/s north along a highway. Inside, a dog moves at 1.75 m/s at an angle 35.0° east of north. What is the velocity of the dog relative to the road?
Answer: 26.4 m/s at 2.17° east of north
You might want to work through Problem F in the text before having students work on this problem.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
For this problem, vpg = vpa + vag Since both of these values are provided, it is a simple vector addition with two right-angle vectors. It might be a good idea to use the component method to add these, since it is unusual to have a right angle between the two vectors. It will also reinforce the notion of breaking vectors down into components before recombining them to get a resultant.

11. Now what do you think?
Suppose you are traveling at a constant 80 km/h when a car passes you. This car is traveling at a constant 90 km/h.
How fast is it going, relative to your frame of reference?
How fast is it moving, relative to Earth as a frame of reference?
Does velocity always depend on the frame of reference?
Does acceleration depend on the frame of reference?
The second car is traveling at 10 km/h relative to your frame of reference, but 90 km/h relative to Earth. Students should now realize that velocity is always relative to a reference frame.
The final question provides a way for you to extend the lesson content. Students may say yes because velocity does depend on a frame of reference, and acceleration is related to velocity. Try to start a class discussion about the two cars. Suppose the 90 km/h car speeds up to 110 km/h in 4 seconds. What acceleration would each observer calculate? It is 5 km/h/s for both observers because the CHANGE IN VELOCITY is the same for both. So, acceleration is NOT relative to the frame of reference.