1. Unit 1 Part 5: Relative Velocity
Book Section 3.4

2. Relative Velocity Relative velocity has to do with perspective.
Who is observing the motion?
Ex: To an observer on the side of a highway, 2 cars might both seem to speed by in a blur. But if the two cars have the same velocity, each driver will see the other as being stationary.

3. In One Dimension
Problem Type 1: Objects A and B move independently of each other.
“the velocity of A relative to B” 𝑣 𝐴𝐵 = 𝑣 𝐴 − 𝑣 𝐵
Remember that these are vectors – keep direction in mind!

4. Example 1 – Same Direction
2 cars are traveling on a highway. Car A is traveling 6.0 m/s east, and car B is traveling 4.0 m/s east. What is the velocity of car A relative to B?
𝑣 𝐴𝐵 = 𝑣 𝐴 − 𝑣 𝐵 =6.0 𝑚 𝑠 𝑒𝑎𝑠𝑡 −4.0 𝑚 𝑠 𝑒𝑎𝑠𝑡=2.0 𝑚 𝑠 𝑒𝑎𝑠𝑡
What is the velocity of car B relative to A?
𝑣 𝐵𝐴 = 𝑣 𝐵 − 𝑣 𝐴 =4.0 𝑚 𝑠 𝑒𝑎𝑠𝑡 −6.0 𝑚 𝑠 𝑒𝑎𝑠𝑡=2.0 𝑚 𝑠 𝒘𝒆𝒔𝒕

5. Example 2 – opposite Directions
2 trains are moving on adjacent tracks. Train A is traveling 40 m/s north, and train B is moving 35 m/s south. What is the velocity of train A relative to B?
𝑣 𝐴𝐵 = 𝑣 𝐴 − 𝑣 𝐵 =40 𝑚 𝑠 𝑛𝑜𝑟𝑡ℎ −35 𝑚 𝑠 𝑠𝑜𝑢𝑡ℎ=75 𝑚 𝑠 𝑛𝑜𝑟𝑡ℎ
What is the velocity of train B relative to A?
𝑣 𝐵𝐴 = 𝑣 𝐵 − 𝑣 𝐴 =35 𝑚 𝑠 𝑠𝑜𝑢𝑡ℎ −40 𝑚 𝑠 𝑛𝑜𝑟𝑡ℎ=75 𝑚 𝑠 𝑠𝑜𝑢𝑡ℎ

6. In One Dimension
Problem Type 2: Object A’s motion depends on the Object B
“the velocity of the A relative to the ground”
𝑣 𝐴𝐺 = 𝑣 𝐴𝐵 + 𝑣 𝐵𝐺

7. Example
A 100-meter boat is moving at 5.2 m/s east. A person, starting from the front of the boat, walks toward the back of the boat at a speed of 2.3 m/s. What is the person’s velocity relative to the shore?
𝑣 𝑃𝑆 = 𝑣 𝑃𝐵 + 𝑣 𝐵𝑆 =2.3 𝑚 𝑠 𝑤𝑒𝑠𝑡+5.2 𝑚 𝑠 𝑒𝑎𝑠𝑡=2.9 𝑚 𝑠 𝑒𝑎𝑠𝑡
How does the problem change if the person is walking the other way on the boat?
𝑣 𝑃𝑆 = 𝑣 𝑃𝐵 + 𝑣 𝐵𝑆 =2.3 𝑚 𝑠 𝑒𝑎𝑠𝑡+5.2 𝑚 𝑠 𝑒𝑎𝑠𝑡=7.5 𝑚 𝑠 𝑒𝑎𝑠𝑡

9. Relative Velocity in 2D

10. WARM UP
A boat travels at a constant speed of 3 m/s on a river. The river’s current has a velocity of 2 m/s east.
If the boat is traveling east in the river, what is the velocity of the boat relative to the shore?
𝑣 𝐵𝑆 = 𝑣 𝐵𝑊 + 𝑣 𝑊𝑆 =3 𝑚 𝑠 𝑒𝑎𝑠𝑡+2 𝑚 𝑠 𝑒𝑎𝑠𝑡=5 𝑚 𝑠 𝑒𝑎𝑠𝑡
If the boat is traveling west in the river, what is the velocity of the boat relative to the shore?
𝑣 𝐵𝑆 = 𝑣 𝐵𝑊 + 𝑣 𝑊𝑆 =3 𝑚 𝑠 𝑤𝑒𝑠𝑡+2 𝑚 𝑠 𝑒𝑎𝑠𝑡=3 𝑚 𝑠 𝑤𝑒𝑠𝑡

11. Suppose the boat is now trying to cross the river, which is 600 meters across. It starts on the south shore and heads straight across.
What is the boat’s velocity (magnitude and direction) relative to the shore?
𝑣 𝐵𝑆 = 𝑣 𝐵𝑊 + 𝑣 𝑊𝑆 =3 𝑚 𝑠 𝑛𝑜𝑟𝑡ℎ+2 𝑚 𝑠 𝑒𝑎𝑠𝑡
𝑣= ≈3.6 𝑚 𝑠
𝜃= tan − ≈33.7° east of north

12. How long will it take the boat to reach the north shore?
𝑡= ∆𝑦 𝑣 𝑦 = =200 𝑠
How far downstream will the boat land?
𝑥= 𝑣 𝑥∗ ∗𝑡=200∗2=400 𝑚

13. Now suppose that the boat must land straight across from where it started.
How far upstream (at what angle) should the boat aim?
𝜃= sin − ≈41.8° west of north
How long will it take the boat to reach the other side?
𝑣 𝑦 = − 2 2 ≈2.2 𝑚 𝑠 so 𝑡= 𝑦 𝑣 𝑦 = ≈273 𝑠

14. Why does it take longer for the boat to cross the river when it aims upstream?
Because in the first scenario, the boat is using all of its speed to get across. In the second, a lot of its speed is spent fighting the current.

15. What is the boat’s velocity (magnitude and direction) relative to the shore?
2.2 m/s north