A Train Is Traveling at 30m/s Relative to the Ground

three.4. Relative Velocity

Midair refueling offers an interesting example of relative velocity. To refuel, the lower plane matches its velocity to that of the tanker (the larger aircraft) and couples to the tanker's delivery tube. During refueling, the relative velocity of the two planes is zero. (© George Hall/Corbis Images)

To someone hitchhiking forth a highway, two cars speeding by in adjacent lanes seem similar a blur. But if the cars have the same velocity, each driver sees the other remaining in identify, 1 lane away. The hitchhiker observes a velocity of perhaps 30 chiliad/s, but each driver observes the other's velocity to be nothing. Clearly, the velocity of an object is relative to the observer who is making the measurement.

Figure 3.16 illustrates the concept of relative velocity by showing a passenger walking toward the front end of a moving railroad train. The people sitting on the train see the passenger walking with a velocity of +2.0 thou/southward, where the plus sign denotes a direction to the right. Suppose the train is moving with a velocity of +9.0 grand/s relative to an observer standing on the ground. Then the ground-based observer would run across the passenger moving with a velocity of +xi m/south, due in part to the walking move and in part to the train'south move. Every bit an aid in describing relative velocity, let usa define the following symbols:

In terms of these symbols, the situation in Figure iii.sixteen tin can be summarized every bit follows:

According to Equation 3.7, v _PG is the vector sum of five _PT and 5 _TG, and this sum is shown in the drawing. Had the passenger been walking toward the rear of the train, rather than the forepart, the velocity relative to the footing-based observer would have been .

The velocity of the passenger relative to the ground-based observer is vPG. It is the vector sum of the velocity vPT of the passenger relative to the train and the velocity vTG of the train relative to the ground:vPGvPTvTG.

Each velocity symbol in Equation three.7 contains a two-letter subscript. The first letter in the subscript refers to the body that is moving, while the 2nd letter indicates the object relative to which the velocity is measured. For example, v _TG and v _PG are the velocities of the Train and Passenger measured relative to the Gcircular. Similarly, 5 _PT is the velocity of the Passenger measured by an observer sitting on the Train.

The ordering of the subscript symbols in Equation 3.seven follows a definite pattern. The starting time subscript (P) on the left side of the equation is likewise the first subscript on the right side of the equation. Likewise, the terminal subscript (G) on the left side is also the terminal subscript on the right side. The third subscript (T) appears only on the correct side of the equation equally the ii "inner" subscripts. The colored boxes below emphasize the pattern of the symbols in the subscripts:

In other situations, the subscripts volition non necessarily exist P, Thou, and T, simply will be compatible with the names of the objects involved in the movement.

Check Your Understanding four

Three cars, A, B, and C, are moving along a direct section of a highway. The velocity of A relative to B is v _AB, the velocity of A relative to C is v _AC, and the velocity of C relative to B is 5 _CB. Fill in the missing velocities in the table.

	v_AB	5_AC	v_CB

one.	?	+twoscore k/due south	+30 m/south
2.	?	+l g/s	–20 m/s
3.	+60 m/s	+twenty thousand/s	?
iv.	–50 grand/s	?	+ten m/s

Groundwork: The relative velocities of three (or more) objects are related past ways of vector addition. Consider how the subscripts are ordered in this addition.

For similar questions (including calculational counterparts), consult Self-Assessment Test 3.2. The test is described at the finish of this section.

(a) A boat with its engine turned off is carried along by the current. (b) With the engine turned on, the boat moves across the river in a diagonal fashion.

Figure 3.17 (a) A boat with its engine turned off is carried along by the current. (b) With the engine turned on, the boat moves beyond the river in a diagonal fashion.

Equation 3.7 has been presented in connectedness with one-dimensional move, simply the upshot is too valid for two-dimensional motion. Figure three.17 depicts a common situation that deals with relative velocity in two dimensions. Part a of the drawing shows a boat beingness carried downstream by a river; the engine of the boat is turned off. In part b, the engine has been turned on, and at present the boat moves beyond the river in a diagonal way because of the combined motility produced past the electric current and the engine. The list below gives the velocities for this type of motion and the objects relative to which they are measured:

The velocity 5 _BW of the boat relative to the water is the velocity measured by an observer who, for instance, is floating on an inner tube and globe-trotting downstream with the current. When the engine is turned off, the boat also drifts downstream with the electric current, and 5 _BW is zero. When the engine is turned on, however, the boat can move relative to the h2o, and v _BW is no longer zero. The velocity five _WS of the water relative to the shore is the velocity of the current measured by an observer on the shore. The velocity v _BS of the boat relative to the shore is due to the combined motion of the boat relative to the water and the motion of the water relative to the shore. In symbols,

The ordering of the subscripts in this equation is identical to that in Equation three.7, although the letters have been changed to reflect a unlike physical situation. Instance ten illustrates the concept of relative velocity in two dimensions.

Example 10Crossing a River

The engine of a gunkhole drives it across a river that is 1800 m wide. The velocity five _BW of the boat relative to the water is 4.0 chiliad/due south, directed perpendicular to the electric current, equally in Effigy 3.18. The velocity v _WS of the water relative to the shore is ii.0 m/south. (a) What is the velocity v _BS of the boat relative to the shore? (b) How long does it take for the boat to cross the river?

The velocity of the boat relative to the shore is vBS. It is the vector sum of the velocity vBW of the boat relative to the water and the velocity vWS of the water relative to the shore:vBSvBWvWS.

Reasoning

Solution

(a)	Since the vectors v _BW and v _WS are perpendicular (see Figure 3.18), the magnitude of five _BS can be adamant by using the Pythagorean theorem: Thus, the boat moves at a speed of 4.5 m/south with respect to an observer on shore. The direction of the boat relative to the shore is given by the bending q in the cartoon:
(b)	The time t for the boat to cantankerous the river is

Occasionally, situations ascend when two vehicles are in relative motion, and it is useful to know the relative velocity of one with respect to the other. Case 11 considers this type of relative movement.

Concept Simulation 3.three

This simulation illustrates the concept of relative velocity by considering a gunkhole traveling across a flowing river. You lot can alter the speed and direction of the boat relative to the h2o, equally well as the velocity of the h2o. The simulation then shows the velocity of the boat as viewed past a person continuing on the shore.

Example xiApproaching an Intersection

Figure 3.19a shows two cars approaching an intersection forth perpendicular roads. The cars have the following velocities:

Find the magnitude and management of five _AB, where

Two cars are approaching an intersection along perpendicular roads.

Figure iii.19 Ii cars are approaching an intersection along perpendicular roads.

Reasoning To find v _AB, we utilise an equation whose subscripts follow the order outlined earlier. Thus,

In this equation, the term v _GB is the velocity of the ground relative to a rider in car B, rather than v _BG, which is given as 15.viii one thousand/south, northward. In other words, the subscripts are reversed. All the same, v _GB is related to v _BG according to This relationship reflects the fact that a passenger in auto B, moving n relative to the basis, looks out the car window and sees the footing moving southward, in the opposite management. Therefore, the equation five _AB

v _AG+5 _GB may exist used to notice v _AB, provided we recognize v _GB as a vector that points contrary to the given velocity v _BG. With this in mind, Effigy 3.xix illustrates how five _AG and v _GB are added vectorially to give v _AB.

Solution From the vector triangle in Figure iii.19b, the magnitude and direction of v _AB tin exist calculated as

and

While driving a car, have you always noticed that the rear window sometimes remains dry, even though rain is falling? This phenomenon is a upshot of relative velocity, as Figure 3.20 helps to explain. Role a shows a machine traveling horizontally with a velocity of v _CG and a raindrop falling vertically with a velocity of v _RG. Both velocities are measured relative to the basis. To determine whether the raindrop hits the window, however, we need to consider the velocity of the raindrop relative to the car, non the basis. This velocity is 5 _RC, and we know that

Here, we have used the fact that . Part b of the drawing shows the tail-to-head arrangement respective to this vector addition and indicates that the direction of 5 _RC is given by the angle q _R. In comparison, the rear window is inclined at an bending q _W with respect to the vertical (see the blowup in part a). When q _R is greater than q _West, the raindrop will miss the window. Notwithstanding, q _R is determined by the speed 5 _RG of the raindrop and the speed 5 _CG of the auto, co-ordinate to . At higher car speeds, the angle q _R becomes as well large for the drop to hitting the window. At a high plenty speed, so, the car only drives out from nether each falling drop