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Wize University Physics Textbook (Master) > Motion in 1/2/3D: Kinematics

Relative Velocity

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Relative Velocity


Velocities are always measured relative to a reference frame. This reference frame could be a moving car or someone standing still in street. The concept of relative velocity lets us consider speeds from different frames of reference. We use vectors representing the velocity of the reference frame.


  • The relative velocity of an object A with respect to another object B is the velocity of object A from the point of view of someone sitting on object B. This velocity is usually shown by v⃗A/B\vec{v}_{A/B}.

  • Relative velocity is essentially the vector difference of the two velocities v⃗A\vec{v}_A and v⃗B\vec{v}_Bwhere these two velocities are measured from a same reference frame such as ground.

vA/B⃗=vA⃗−vB⃗\boxed{\vec{v_{A/B}}=\vec{v_A}-\vec{v_B}}










Wize Tip
Remember that velocities are vectors. So, we will have to use vector addition and subtraction when working with above equation.


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General strategy

  1. Write out your velocities with the subscripts in the order of object-respect to-frame of reference.
  2. Write out the relation between them using the above equation.
  3. Sometimes we can draw a diagram showing velocity vectors to guess what the relationship between the velocities described in the problem are
  4. Solve for what you want.
Exam Tip
Usually problems involving a boat, swimmer crossing a river, or air plane flying in a windy condition could be solved using this method!



















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Example: Relative Velocity of a Ship


A ship sets sail from Rotterdam, heading due north at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a direction 40.0° north of east. What is the velocity of the ship relative to Earth?

Solution:

It is good to draw a diagram and show velocities given in this problem and label them properly:

V⃗SE=V⃗SW+V⃗WE\vec{V}_{SE} = \vec{V}_{SW}+\vec{V}_{WE}

We can write down above vector equation in terms of its components:

(VSE)x=(VSW)x+(VWE)x({{V}_{SE}})_x = ({{V}_{SW}})_x+({V}_{WE})_x
(VSE)x=0+(1.50)cos(40)=1.15m/s({{V}_{SE}})_x= 0+ (1.50)cos(40)= 1.15m/s

(VSE)y=(VSW)y+(VWE)y({{V}_{SE}})_y = ({{V}_{SW}})_y+({V}_{WE})_y
(VSE)y=7+(1.50)sin(40)=7.96m/s({{V}_{SE}})_y = 7+ (1.50) sin (40)=7.96m/s

To find the speed of ship respect to earth we have to find the magnitude of velocity vector

VSE=(VSE)x2+(VSE)y2{{V}_{SE}} = \sqrt{({{V}_{SE}})_x^2+({{V}_{SE}})_y^2}
VSE=(1.14)2+(7.96)2=8.04m/s{{V}_{SE}} = \sqrt{(1.14)^2+ (7.96)^2}= 8.04m/s

In addition, the direction of velocity vector could be found using below equation:

tan(θ)=(VSE)y(VSE)x=7.961.15tan(\theta) = \dfrac{(V_{SE})_y}{(V_{SE})_x}= \dfrac{7.96}{1.15}

θ=81.78°\theta= 81.78\degree

This angle is measured counter-clockwise respect to positive x-axis. So, this vector is 81.78 degrees north of east.

A pilot in a Boeing 747 is beginning a 1500km flight. The plane's speed is 1000km/h and air traffic control says that he will have to head 15o15^owest of south to maintain a southward course. If the flight takes 100min, what is the wind speed?
Extra Practice
Relative velocity
Dave swims at a constant 2.5 m/s across a river flowing at 2 m/s. What is the:
(a) the relative velocity of Dave, and
(b) what is Dave's direction of travel, with respect to the ground?
In an elevator going up at constant speed, you drop your wallet.
a) Will it hit the ground faster, slower, or in the same amount of time compared to a situation where the elevator was at rest? Explain your answer.
b) What if the elevator was initially at rest and started moving while your wallet was falling to the ground? Explain your answer.
Andy is trying to catch a bus, but the bus is pulling away already at 40.0o from +x axis at 5.0m/s. Bob sitting in the bus sees Andy running at 48.6o below the -x axis with a speed of 4.28m/s.
How fast was Andy running and in what direction?
Relative velocities example
Say you are on the ground, outside, controlling a drone video camera. In your video, it seems like a bird flew by at 5.0 m/s southwest. From your point of view on the ground, the bird flew 1.0 m/s northwest. What is the velocity of your drone relative to you on the ground? Give your answer in terms of magnitude and the angle relative to the +x axis.
Franklin is coming over to your house. He first travels at 3m/s East, and then he travels 3m/s South. What direction is his average acceleration?
Turning acceleration
Joe is running 10m/s to the east. And then he's running 10m/s to the south. What direction is his average acceleration?
Practice Example: Relative Velocity
A pilot in a Boeing 747 is beginning a 1500km flight. The plane's speed is 1000km/h and air traffic control says that he will have to head 15o15^owest of south to maintain a southward course. If the flight takes 100min, what is the wind speed?
Relative velocity
A duck swims at a constant speed from one side of a river to the other side in a time of 4 seconds. The river is 6 meters wide and it is flowing at a speed of 2 m/s. What is the
(a) the velocity of the duck and
(b) what is its direction of travel, with respect to the ground?
Practice: Relative Velocity 2
A train travels at 50 m/s to north with respect to the ground. A boy on the train runs at 10m/s to the south with respect to the train. Find the velocity of the boy with respect to the ground.
I want to direct a remote-control toy boat due west across a river, but my jerk friend Dave broke the rudder, so I can’t turn mid-journey. The boat travels at a speed of 8 m/s, and the river is flowing south at 5 m/s. What direction should I point my boat?
A pilot in a Boeing 747 begins a 1500-km flight. The plane's speed is 1000 km/h, and air traffic control says you'll have to head 15o west of south to maintain a southward course. If the flight takes 100 min, what's the wind velocity?
A pilot in a Boeing 747 is beginning a 1500 km flight. The plane's speed is 1000 km/h and air traffic control says that he will have to head 15o west of south to maintain a southward course. If the flight takes 100 minutes, what is the wind velocity? (Enter your answer to three sig figs, in units of km/h.)