RELATIVE VELOCITY

The velocity of an object with respect to another object.

Notation• vA = velocity of object A with respect to astationary object.• vAG= velocity of object A with respect to

ground (stationary object on the ground)• vA and VAG have the same meaning.• vAB=velocity of object A in the frame of

reference (with respect to) object B• vAB=vA-vB (velocity of object A with respect to

ground - velocity of object B with respect to ground.)

Relative Velocity (One Dimension)

A

B

vA=+5.0 m/s vB=0 m/s vAB=vA-vB =+5.0 m/s – 0 m/s = +5.0 m/s

vBA=vB-vA = 0-(+5.0m/s) = -5.0 m/s

vAB = -vBA (Always)

Relative Velocity (Two Objects Moving in Opposite Directions)

• The relative velocity of two objects moving in opposite directions is the sum of the two speeds with the appropriate frame of reference direction.

A

B

vA=-10 m/s vB=+5.0 m/s

vAB=vA-vB = -10 m/s – (+5.0 m/s) = -15.0 m/svBA = +15.0 m/s

Relative Velocity (Two Objects Moving in the Same Direction)

• The relative velocities of two objects moving in the same direction is the difference of the two speeds and with the appropriate frame of reference direction.

A

B

vA=+10 m/s vB=+5.0 m/s

vAB=vA-vB = +10 m/s – (+5.0 m/s) = +5.0 m/svBA = -5.0 m/s

Relative Velocity(One object moving with another object)

vBC=+20 m/s vCG=+10 m/s

vBG=vBC+vCG

vBC =velocity of ball with respect to carvCG=velocity of the car with respect to groundvBG=velocity of ball with respect to ground

vBG=+20m/s +(+10 m/s)= +30 m/s

Check of Relative Velocities

• vBG=vBC+vCG

• vBG=(vB-vC)+(vC-vG)=vB-VG

Relative Velocity (Two Dimensions)

AB

θA

θB

vABx=vAcosθA-vBcosθB vABy=vAsinθA-vBsinθB

vA

vB

2AB

2ABAB )v()v(v

yx

x

y

AB

AB1

v

vtan

Situation 1:

`

Relative Velocity in Two Dimensions

vWG

vBW

vBW

vWG

vBG

vBG=vBW+vWG2WG

2BWBG vvv

θ

θ=tan-1(vWG/vBW)

What is the velocity of the boat in the earth’s frame of reference?

Situation 2:

vWG= velocity of water with respect to ground vBW= velocity of boat with respect to water

vBG= velocity of boat with respect to the ground.

Relative Velocity in Two DimensionsvWG

vBW

vBW

vWG

vBG=vR

vBG=vBW+vWG2WG

2BWRBG vvvv

θ

θ=sin-1(vWG/vBW)

The angle needed to travel directly across the stream.

Situation 3:

vR

vR = resultant boat speed

vWG

vB

vBG=vB + vW = vR

θ=tan-1(x/d) where θ=α+φ, which is the resultant angle of the boat.α= boat angle, φ=angle between boat’s aimed direction (boat angle) and actual direction.

α

The angle necessary to dock a specific distance downstream:x

d

φ

What angle, α, must the boat be directed to dock a distance, x, downstream while crossing a river that is a distance, d, wide? Known: vW,,vB, x, and d.

θ

vW

Situation4:

vW

vB

α

x

d

φ

θ

β

vBGvBG=vR

z

z

Use law of sines.

vW/vB=sin φ/sin z, solve for φ

α = θ - φ

vR = resultant velocity from water and boat

Situation 4 continued

vW

vB

α

x

d

φ θ

vBGvBG=vR

vR = resultant velocity from water and boat

Distance to dock upstream.

Situation 4a

vWG

vB

α

x

d

φ

θ

vBGvBG=vR

z

z

vR = resultant velocity from water and boat

Distance downstream docked based on specific boat direction:

Known: vWG, vBW, d, and α.How far will the boat aimed at an angle, α, dock downstream?

Situation 5

vW

vB

α

x

d

φ

θ

vBG

vBG=vR

z

z

vR = resultant velocity from water and boat

vRx=vW+vB(sin α) vRy = vB(cos α)

= vBx

2Ry

2RxR )V()(VV

tan θ=(x/d) solve for x. θ=tan-1(vRx/vRy)

Situation 5 continued

vBx