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3-6. Addition of Velocities

Learning Objectives

  • Apply principles of vector addition to determine relative velocity.
  • Explain the significance of the observer in the measurement of velocity.

Relative Velocity

If a person rows a boat across a rapidly flowing river and tries to head directly for the other shore, the boat instead moves diagonally relative to the shore, as in Figure 1. The river carries the boat downstream. Similarly, if a small airplane flies overhead in a strong crosswind, it does not move in the direction in which it is pointed because the air mass carries it sideways.

Figure 1: A boat trying to head straight across a river will actually move diagonally relative to the shore. Its total velocity relative to the shore is the sum of its velocity relative to the river plus the velocity of the river relative to the shore.
Figure 2: An airplane heading straight north is carried west and slowed by wind. Its total velocity is the sum of its airspeed and the wind velocity.

In each case, the object has a velocity relative to a medium (river or air), and that medium has a velocity relative to the observer on the ground. The total velocity is the vector sum of these two velocities.

Adding Velocities

Velocity is a vector quantity—it has both magnitude and direction. In one-dimensional motion, velocities add like numbers. In two dimensions, analytical or graphical methods are used. The relationships between velocity magnitude (v) and its components (vx, vy) are:

vₓ = v cos θ
vᵧ = v sin θ
v = √(vₓ² + vᵧ²)
θ = tan⁻¹(vᵧ / vₓ)
Figure 3: The velocity of an object traveling at an angle θ to the horizontal is the sum of component vectors vₓ and vᵧ.

Take-Home Experiment: Relative Velocity of a Boat

Fill a bathtub half-full of water. Place a toy boat on the surface and unplug the drain so water starts to flow. Try pushing the boat straight across. Observe how the flow affects its path and compare the directions of flow, heading, and actual motion.

Example 1: A Boat on a River

Figure 4: A boat travels across a river at 0.75 m/s while the current flows at 1.20 m/s. Calculate its total velocity relative to the shore.

Solution:
Using Pythagorean addition:
vtot = √(1.20² + 0.75²) = 1.42 m/s
and θ = tan⁻¹(0.75 / 1.20) = 32°.
The boat moves diagonally downstream with a total velocity of 1.42 m/s at an angle of 32°.

Example 2: Wind Velocity Causes an Airplane to Drift

Figure 5: An airplane moves north at 45.0 m/s, but wind changes its total velocity to 38.0 m/s, 20° west of north. Find the wind’s velocity.

Solution:
Breaking into components gives:
vwx = −13.0 m/s (west) and vwy = −9.29 m/s (south),
resulting in vw = 16.0 m/s at θ = 35.6°.

Relative Velocities and Classical Relativity

Velocity depends on the observer’s frame of reference. Relative velocity describes how an object’s motion is measured differently by observers in different reference frames. Classical relativity, introduced by Galileo and Newton, applies when speeds are much less than 1% of the speed of light.

Figure 6: Classical relativity. A sailor and a shore observer see the same motion differently but both agree where the object lands.

Example 3: Airline Passenger Drops a Coin

Figure 7: A coin dropped in a moving airplane appears to fall straight down inside but follows a curved path relative to the ground.

Solution:
(a) Relative to the plane: vy = −5.42 m/s (straight down).
(b) Relative to the Earth: v = √(260² + 5.42²) = 260.06 m/s at θ = −1.19°.
Observers in the plane and on the ground see different paths due to different frames of reference.

Making Connections: Relativity and Einstein

Einstein’s theory of relativity extended classical relativity by showing that the speed of light is constant for all observers. It led to new insights: time varies by observer, energy and mass are equivalent, and space-time relationships reshape our understanding of motion.

PhET Exploration: Motion in 2D

Figure 8: Explore 2D motion using interactive simulations to visualize velocity and acceleration vectors.

Summary

Velocities in two dimensions follow vector addition principles:

vₓ = v cos θ
vᵧ = v sin θ
v = √(vₓ² + vᵧ²)
θ = tan⁻¹(vᵧ / vₓ)

Relative velocity depends on the observer’s frame of reference. Classical relativity describes these effects for speeds below 1% of the speed of light.

Conceptual Questions

  1. What frame of reference do you use when driving a car or flying in an airplane?
  2. Why doesn’t a basketball player need to watch the ball while dribbling?
  3. Under what condition would a ball thrown backward from a moving truck fall straight down to a roadside observer?
  4. Describe the path of a hat falling off a jogger’s head as seen by the jogger and by a stationary observer.
  5. When dirt falls from a moving truck, how does its velocity compare relative to the truck and the ground?

Problems & Exercises

  1. Bryan Allen pedaled a human-powered aircraft across the English Channel. Find his total displacement, average velocity relative to the air, and total displacement relative to the air mass. Solution: (a) 35.8 km; (b) 5.53 m/s; (c) 56.1 km.
  2. A seagull flies 6.00 km against the wind at 9.00 m/s. Find the wind’s velocity and the total travel time for the round trip.
  3. Two marathon runners are 45 m apart. One runs 3.50 m/s, the other 4.20 m/s. Who wins? Solution: Second runner wins, 4.17 m ahead.
  4. Verify that the dropped coin from Example 3 travels 144 m horizontally while falling 1.50 m.
  5. A quarterback moves backward at 2.00 m/s and throws a football 18.0 m downfield at 25°. Find the initial velocity relative to him. Solution: 17.0 m/s at 22.1°.

Additional problems explore ship and airplane velocities, river currents, galaxy expansion (Figure 9), and real-world vector addition cases.

Figure 9: Five galaxies at increasing distances from the Milky Way show proportional velocities, illustrating the expanding universe.

Summary Insight: Understanding velocity addition and relativity enhances problem-solving in physics, navigation, and astronomy. Choosing the correct reference frame simplifies calculations and explains why different observers measure different motion paths.