Relative Velocities Assignment

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One of the most confusing concepts for young aerodynamicists is the relative velocity between objects. Aerodynamic forces are generated by an object moving through a fluid (liquid or gas). A fixed object in a still fluid does not generate aerodynamic forces. (Hot air balloons "lift" because of buoyancy forces, not aerodynamic forces.) To generate lift, an object must move through the air, or air must move past the object. [Some aircraft (like the Harrier) use engine thrust to "lift" the vehicle. But this, again, is not aerodynamic lift.] Aerodynamic lift depends on the square of the velocity between the object and the air. Now things get confusing because not only can the object be moved through the air, but the air itself can move. To properly define the velocity, it is necessary to pick a fixed reference point and measure velocities relative to the fixed point. In this slide, the reference point is fixed to the ground, but it could just as easily be fixed to the aircraft. It is important to understand the relationships of wind speed to ground speed to airspeed.

Wind Speed

For a reference point picked on the ground, the air moves relative to the reference point at the wind speed. Notice that the wind speed is a vector quantity and has both a magnitude and a direction. (A 20 mph wind from the west is different from a 20 mph wind from the east.) The wind has components in all three primary directions (north-south, east-west, and up-down). In this figure, we are considering only velocities along the aircraft's flight path. And a positive velocity is defined to be in the direction of the aircraft's motion. We are neglecting cross winds, which occur perpendicular to the flight path but parallel to the ground, and updrafts and downdrafts, which occur perpendicular to the ground.

Ground Speed

The velocity of the object measured relative to the ground is called the ground speed. Again, this is a vector quantity.

Airspeed

The important quantity in the generation of lift is the relative velocity between the object and the air, which is called the airspeed. Airspeed cannot be directly measured from a ground position, but must be computed from the ground speed and the wind speed. Airspeed is the vector difference between the ground speed and the wind speed. On a perfectly still day, the airspeed is equal to the ground speed. But if the wind is blowing in the same direction that the aircraft is moving, the airspeed will be less than the ground speed.

Examples

Suppose we had an airplane that could take off on a windless day at 100 mph (liftoff airspeed is 100 mph). Now suppose we had a day in which the wind was blowing 20 mph towards the west. If the airplane takes off going east, it experiences a 20 mph headwind (wind in your face). Since we have defined a positive velocity to be in the direction of the aircraft's motion, a headwind is a negative velocity. While the plane is sitting still on the runway, it has a ground speed of 0 and an airspeed of 20 mph [airspeed = ground speed (0) - wind speed (-20) ]. At liftoff, the airspeed is 100 mph, the wind speed is -20 mph and the ground speed will be 80 mph [airspeed (100) = ground speed (80) - wind speed (-20) ].

If the plane took off to the west, it would have a 20 mph tailwind (wind at your back). Since the wind and aircraft direction are the same, we assign a "+" to the wind speed. At liftoff, the airspeed is still 100 mph, the wind speed is 20 mph and the ground speed will now be 120 mph [airspeed (100) = ground speed (120) - wind speed (20) ]. So the aircraft will have to travel faster (and farther) along the ground to achieve liftoff conditions with the wind at it's back.

Significance of Understanding Relative Velocity

The importance of the relative velocity explains why airplanes take off and land on different runways on different days. Airplanes always try to take off and land into the wind. This requires a lower ground speed (as we saw in the above example), which means the plane can take off or land in the shortest distance traveled along the ground. Since runways have a fixed length, you want to get airborne as fast as possible on takeoff and stopped as soon as possible on landing. In the old days, a large "wind sock" was hung near the runway for pilots to see which way the wind was blowing to adjust their takeoff and landing directions. Now mechanical or electronic devices provide the information that is radioed to the cockpit.

The relationship between airspeed, wind speed, and ground speed explains why wind tunnel testing is possible and how kites fly.

  • In the wind tunnel, the ground speed is zero. (The object is fixed to the walls of the tunnel.) The airspeed is then the negative of the wind speed that is generated in the tunnel. Whether the object moves through the air, or the air moves over the object, the forces are the same.
  • A kite usually has no ground speed (the kite is held on the end of a string). But the kite still has an airspeed that is equal to the wind speed. (You can fly a kite only with the wind at your back.)


Guided Tours
  • Into the Wind:

Beginner's Guide to Aerodynamics
Beginner's Guide to Propulsion
Beginner's Guide to Model Rockets
Beginner's Guide to Kites
Beginner's Guide to Aeronautics


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byTom Benson
Please send suggestions/corrections to: benson@grc.nasa.gov

 

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 along a highway, two cars speeding by in adjacent lanes seem like a blur. But if the cars have the same velocity, each driver sees the other remaining in place, one lane away. The hitchhiker observes a velocity of perhaps 30 m/s, but each driver observes the other’s velocity to be zero. 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 of a moving train. The people sitting on the train see the passenger walking with a velocity of +2.0 m/s, where the plus sign denotes a direction to the right. Suppose the train is moving with a velocity of +9.0 m/s relative to an observer standing on the ground. Then the ground-based observer would see the passenger moving with a velocity of +11 m/s, due in part to the walking motion and in part to the train’s motion. As an aid in describing relative velocity, let us define the following symbols:

In terms of these symbols, the situation in Figure 3.16 can be summarized as follows:

 (3.7) 

or

According to Equation 3.7, vPG is the vector sum of vPT and vTG, and this sum is shown in the drawing. Had the passenger been walking toward the rear of the train, rather than the front, the velocity relative to the ground-based observer would have been .

Figure 3.16  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:vPG=vPT+vTG.

Each velocity symbol in Equation 3.7 contains a two-letter subscript. The first letter in the subscript refers to the body that is moving, while the second letter indicates the object relative to which the velocity is measured. For example, vTG and vPG are the velocities of the Train and Passenger measured relative to the Ground. Similarly, vPT is the velocity of the Passenger measured by an observer sitting on the Train.

The ordering of the subscript symbols in Equation 3.7 follows a definite pattern. The first subscript (P) on the left side of the equation is also the first subscript on the right side of the equation. Likewise, the last subscript (G) on the left side is also the last subscript on the right side. The third subscript (T) appears only on the right side of the equation as the two “inner” subscripts. The colored boxes below emphasize the pattern of the symbols in the subscripts:

In other situations, the subscripts will not necessarily be P, G, and T, but will be compatible with the names of the objects involved in the motion.



Check Your Understanding 4


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 across the river in a diagonal fashion.

Equation 3.7 has been presented in connection with one-dimensional motion, but the result is also valid for two-dimensional motion. Figure 3.17 depicts a common situation that deals with relative velocity in two dimensions. Part a of the drawing shows a boat being carried downstream by a river; the engine of the boat is turned off. In part b, the engine has been turned on, and now the boat moves across the river in a diagonal fashion because of the combined motion produced by the 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 velocityvBW of the boat relative to the water is the velocity measured by an observer who, for instance, is floating on an inner tube and drifting downstream with the current. When the engine is turned off, the boat also drifts downstream with the current, and vBW is zero. When the engine is turned on, however, the boat can move relative to the water, and vBW is no longer zero. The velocity vWS of the water relative to the shore is the velocity of the current measured by an observer on the shore. The velocity vBS 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 3.7, although the letters have been changed to reflect a different physical situation. Example 10 illustrates the concept of relative velocity in two dimensions.



Example 10  Crossing a River

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

Concept Simulation 3.3

This simulation illustrates the concept of relative velocity by considering a boat traveling across a flowing river. You can change the speed and direction of the boat relative to the water, as well as the velocity of the water. The simulation then shows the velocity of the boat as viewed by a person standing on the shore.



Example 11  Approaching an Intersection

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

Find the magnitude and direction of vAB, where

Figure 3.19  Two cars are approaching an intersection along perpendicular roads.

Reasoning  To find vAB, we use an equation whose subscripts follow the order outlined earlier. Thus,

In this equation, the term vGB is the velocity of the ground relative to a passenger in car B, rather than vBG, which is given as 15.8 m/s, northward. In other words, the subscripts are reversed. However, vGB is related to vBG according to
This relationship reflects the fact that a passenger in car B, moving northward relative to the ground, looks out the car window and sees the ground moving southward, in the opposite direction. Therefore, the equation vAB=vAG+vGB may be used to find vAB, provided we recognize vGB as a vector that points opposite to the given velocityvBG. With this in mind, Figure 3.19 illustrates how vAG and vGB are added vectorially to give vAB.
Problem solving insight
In general, the velocity of object R relative to object S is always the negative of the velocity of object S relative to R:vRS=–vSR

Solution From the vector triangle in Figure 3.19b, the magnitude and direction of vAB can be calculated as

and

While driving a car, have you ever noticed that the rear window sometimes remains dry, even though rain is falling? This phenomenon is a consequence of relative velocity, as Figure 3.20 helps to explain. Part a shows a car traveling horizontally with a velocity of vCG and a raindrop falling vertically with a velocity of vRG. Both velocities are measured relative to the ground. To determine whether the raindrop hits the window, however, we need to consider the velocity of the raindrop relative to the car, not the ground. This velocity is vRC, and we know that

Here, we have used the fact that . Part b of the drawing shows the tail-to-head arrangement corresponding to this vector addition and indicates that the direction of vRC is given by the angle R. In comparison, the rear window is inclined at an angle W with respect to the vertical (see the blowup in part a). When R is greater than W, the raindrop will miss the window. However, R is determined by the speed vRG of the raindrop and the speed vCG of the car, according to . At higher car speeds, the angle R becomes too large for the drop to hit the window. At a high enough speed, then, the car simply drives out from under each falling drop

Figure 3.20  (a) With respect to the ground, a car is traveling at a velocity of vCG and a raindrop is falling at a velocity of vRG. The rear window of the car is inclined at an angle W with respect to the vertical. (b) This tail-to-head arrangement of vectors corresponds to the equation vRC=vRG–vCG.


Self-Assessment Test 3.2

Test your understanding of the key ideas in Section 3.4:

· Relative Velocity  · Vector Addition of Relative Velocities





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