All motion is relative; that is, motion must be measured relative to a frame of reference. For example, if you’re sitting in a rowboat that is floating down a river with the current, your friend sitting on the shore may see you moving downstream with a velocity of 10 m/s [W]. Relative to your friend’s frame of reference, you have a velocity of 10 m/s [W]. On the other hand, relative to a passenger sitting in your boat, you have a velocity of zero; you’re not moving because you are both at rest in your common frame of reference, the boat. Relative to your frame of reference in the boat, your friend on the shore is moving at 10 m/s [E]. This example is a simple one dimensional example of relative motion.

Two dimensional relative motion Relative Velocity Problems All velocity is measured from a reference frame (or point of view). Velocity with respect to a reference frame is called relative velocity. A relative velocity has two subscripts, one for the object, the other for the reference frame. Relative velocity problems relate the motion of an object in two different reference frames. refers to the object refers to the reference frame velocity of object a relative to reference frame b velocity of reference frame b relative to reference frame c velocity of object a relative to reference frame c

Relative Velocity vpg = velocity of person relative to ground At the airport, if you walk on a moving sidewalk, your velocity is increased by the motion of you and the moving sidewalk. vpg = velocity of person relative to ground vps = velocity of person relative to sidewalk vsg = velocity of sidewalk relative to ground When flying against a headwind, the plane’s “ground speed” accounts for the velocity of the plane and the velocity of the air. vpe = velocity of plane relative to earth vpa = velocity of plane relative to air vae = velocity of air relative to earth

The equation that relates these three velocities is We will examine a number of examples where an object is moving through a medium, like air or water, which is in turn moving relative to Earth or the ground. In order to keep these velocities distinct, we will use a series of subscripts. The equation that relates these three velocities is

A river-crossing problem: Part A A physics student wants to cross the Bernoulli River. He hops into his bass boat at A-ville and drives straight, due north, toward B-ville with a velocity of 5.0 km/h. If the river is 5.0 km wide, how long does it take the teacher to reach the other side?

Solution and Connection to Theory Since the boat isn’t accelerating, we can use the defining equation for speed. It takes 1.0 h to reach B-ville.

A river-crossing problem: Part B Let’s introduce a current, flowing at 2.0 km/h [E], which will prevent the boat from landing at B-ville by pushing it farther east. Instead, the boat will land at C-ville, as shown in Figure below. How does the current affect the time required to cross the river? Does the boat take the same amount of time, a shorter period of time, or a longer period of time?

Solution and Connection to Theory It will take the boat exactly the same amount of time to reach the other shore, regardless of whether a current is present or not. Time isn’t affected because the boat’s velocity [N] and the current’s velocity [E] are perpendicular to each other. Therefore, the boat’s velocity has no components in the same (or in the opposite) direction as the current’s velocity and vice versa. The two velocities therefore have no effect on each other. Since the boat is travelling at the same velocity due north as in the Part A of this example, and the distance across the river hasn’t changed, it will take the same amount of time to cross the river.

A river-crossing problem: Part C Given that it still takes 1.0 h to reach the other shore, how far is it from B-ville to C-ville?

Solution and Connection to Theory Q: The only velocity causing the boat to move downstream from B-ville to C-ville is the river current’s velocity. Therefore, we can calculate the distance by using the defining equation for speed and substituting the current’s speed, represented by the subscript cg: Therefore, the distance between B-ville and C-ville is 2.0 km.

Q: what is the ground velocity vbg, of the boat; that is, the velocity of the boat relative to a person standing on the ground (or shore). To solve the problem, we add the two perpendicular vectors, vbc (the velocity of the boat relative to the current) and vcg (the velocity of the current relative to the ground) using Pythagoras’ theorem and the tangent function. Solution:

Problems Involving Non-perpendicular Vectors: A boat-navigation problem: The physics student from Example 1Part B wants to go to B-ville, which is directly north of A-ville. To do so, the bass boat must be aimed upstream to compensate for the current, as shown in Figure. The current velocity is 2.0 km/h [E] and the boat’s speed is 5.0 km/h. a) In which direction must the boat be pointed in order to land at B-ville? b) What is the ground velocity of the boat? c) How long will it take the boat to reach the other shore?

Solution and Connection to Theory Let vcg be the velocity of the current relative to the ground, let vbc be the velocity of the boat relative to the water, and let vbg be the velocity of the boat relative to the ground. The triangle in Figure below is a right-angle triangle. Therefore, we can determine the direction in which the boat must be pointed by using the cosine function: In Figure, vbc shows the direction in which the boat must be pointed in order to land at B-ville; that is, [W66°N] or [N24°W].

b) To determine the magnitude of the ground velocity, vbg,

Relative Velocity vpe = velocity of plane relative to earth When flying with a crosswind, the plane’s “ground speed” is the resultant of the velocity of the plane and the velocity of the air. vpe = velocity of plane relative to earth vpa = velocity of plane relative to air vae = velocity of air relative to earth Pilots must fly with crosswind but not be sent off course. Sometimes the vector sums are more complicated!

Relative Motion Our understanding of relative motion has many applications. Consider the motion of a boat across a river. Usually a captain wants to arrive at a specific point on the other side. Once disconnected from the shore, the boat will move in the reference frame of the river. The boat will need to head into the current in order to arrive at its destination.

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