What Is the Speed of the Boat, in M/s, With Respect to Earth?

Learning Objectives

By the terminate of this section, you will be able to:

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

Relative Velocity

If a person rows a boat across a apace flowing river and tries to head directly for the other shore, the boat instead moves diagonally relative to the shore, as in Figure i. The gunkhole does non move in the direction in which it is pointed. The reason, of class, is that the river carries the boat downstream. Similarly, if a minor airplane flies overhead in a strong crosswind, yous can sometimes run into that the plane is not moving in the direction in which it is pointed, as illustrated in Figure 2. The plane is moving directly ahead relative to the air, but the movement of the air mass relative to the ground carries it sideways.

Figure 1. A gunkhole trying to head straight across a river will actually move diagonally relative to the shore as shown. Its full velocity (solid arrow) 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 ii. An aeroplane heading directly north is instead carried to the west and slowed down by current of air. The plane does not move relative to the footing in the direction it points; rather, it moves in the direction of its total velocity (solid arrow).

In each of these situations, an object has a velocity relative to a medium (such as a river) and that medium has a velocity relative to an observer on solid basis. The velocity of the object relative to the observer is the sum of these velocity vectors, equally indicated in Figure ane and Figure 2. These situations are only two of many in which information technology is useful to add together velocities. In this module, we first re-examine how to add velocities and and so consider sure aspects of what relative velocity ways. How practice we add velocities? Velocity is a vector (information technology has both magnitude and management); the rules of vector addition discussed in Vector Addition and Subtraction: Graphical Methods and Vector Improver and Subtraction: Analytical Methods apply to the improver of velocities, just as they do for any other vectors. In one-dimensional motility, the improver of velocities is simple—they add together like ordinary numbers. For instance, if a field hockey actor is moving at v g/south straight toward the goal and drives the ball in the same direction with a velocity of xxx m/s relative to her trunk, then the velocity of the ball is 35 m/s relative to the stationary, profusely sweating goalkeeper standing in forepart of the goal. In two-dimensional motility, either graphical or analytical techniques tin can be used to add velocities. We volition concentrate on analytical techniques. The following equations give the relationships between the magnitude and management of velocity (5 and θ) and its components (v _x and v _y ) forth the x – and y -axes of an appropriately chosen coordinate system:

5 _x = v cosθ

v _y = 5 sinθ

[latex]v=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}}[/latex]

θ = tan ^{− ane} (v _y /5 _x ).

Figure iii. The velocity, v , of an object traveling at an bending θ to the horizontal centrality is the sum of component vectors v ₁₀ and v _y.

These equations are valid for any vectors and are adapted specifically for velocity. The first two equations are used to find the components of a velocity when its magnitude and management are known. The last two are used to find the magnitude and direction of velocity when its components are known.

Take-Home Experiment: Relative Velocity of a Boat

Fill a bathtub half-total of water. Take a toy gunkhole or some other object that floats in h2o. Unplug the bleed so water starts to bleed. Effort pushing the boat from one side of the tub to the other and perpendicular to the flow of water. Which manner practise you demand to push the boat so that it ends up immediately opposite? Compare the directions of the period of water, heading of the boat, and actual velocity of the gunkhole.

Case ane. Adding Velocities: A Gunkhole on a River

Figure 4. A gunkhole attempts to travel directly across a river at a speed 0.75 m/s. The current in the river, all the same, flows at a speed of one.twenty m/southward to the right. What is the total displacement of the boat relative to the shore?

Refer to Figure 4 which shows a boat trying to go straight across the river. Let us calculate the magnitude and management of the boat's velocity relative to an observer on the shore, v_tot. The velocity of the boat, v_boat, is 0.75 m/due south in the y -management relative to the river and the velocity of the river, five_river, is 1.20 m/s to the correct.

Strategy

Nosotros start by choosing a coordinate system with its x-centrality parallel to the velocity of the river, equally shown in Figure 4. Because the boat is directed straight toward the other shore, its velocity relative to the water is parallel to the y-axis and perpendicular to the velocity of the river. Thus, we can add together the two velocities by using the equations [latex]{v}_{\text{tot}}=\sqrt{{{v}_{ten}}^{2}+{{5}_{y}}^{2}}[/latex] and θ= tan⁻¹(5 _y /v _x ) directly.

Solution

The magnitude of the full velocity is

[latex]{v}_{\text{tot}}=\sqrt{{{v}_{10}}^{2}+{{v}_{y}}^{2}}[/latex]

where

v _x = v _river = 1.xx g/south

and

five _y =v _boat= 0.750 chiliad/s.

Thus,

[latex]{v}_{\text{tot}}=\sqrt{({i.twenty}\text{ m/s})^{ii} + ({0.750}\text{ thou/due south})^{two}}[/latex]

yielding

five _tot= i.42 chiliad/south.

The direction of the total velocity θ is given by:

θ= tan⁻¹(v _y /five _x ) = tan^−one(0.750/1.20).

This equation gives

θ= 32.0º.

Give-and-take

Both the magnitude v and the management θ of the total velocity are consistent with Figure 4. Annotation that considering the velocity of the river is big compared with the velocity of the boat, it is swept rapidly downstream. This event is evidenced by the minor angle (only 32.0º) the total velocity has relative to the riverbank.

Example ii. Computing Velocity: Air current Velocity Causes an Aeroplane to Drift

Calculate the current of air velocity for the situation shown in Figure 5. The airplane is known to exist moving at 45.0 m/s due due north relative to the air mass, while its velocity relative to the ground (its full velocity) is 38.0 k/due south in a direction twenty.0º west of n.

Figure five. An airplane is known to be heading north at 45.0 one thousand/due south, though its velocity relative to the footing is 38.0 m/s at an angle west of northward. What is the speed and management of the wind?

Strategy

In this problem, somewhat different from the previous example, we know the total velocity v_tot and that information technology is the sum of 2 other velocities, v_w (the wind) and 5_p (the aeroplane relative to the air mass). The quantity 5_p is known, and we are asked to observe v_w. None of the velocities are perpendicular, only information technology is possible to find their components along a common prepare of perpendicular axes. If we can observe the components of 5_westward, then we can combine them to solve for its magnitude and direction. As shown in Figure 5, we cull a coordinate organization with its x-axis due eastward and its y-axis due northward (parallel to 5_p). (Yous may wish to wait back at the discussion of the add-on of vectors using perpendicular components in Vector Addition and Subtraction: Analytical Methods.)

Solution

Because v_tot is the vector sum of the five_w and v_p, its ten– and y-components are the sums of the x– and y-components of the air current and plane velocities. Note that the plane only has vertical component of velocity so five _p10 = 0 and v _py =five _p. That is,

v _{tot x} = v _{w x}

and

v _toty =v _{due westx} +v _p.

We tin use the first of these two equations to detect five _westx :

v _wx =5 _totx =v _totcos 110º.

Because five _tot = 38.0 m/s and cos 110º=–0.342 we take

five _{due westx} = (38.0 one thousand/s)(–0.342) = –thirteen.0 k/southward.

The minus sign indicates motion due west which is consequent with the diagram. Now, to notice 5 _wy we notation that

v _{tot y} = v _{w x} + v _p

Here v _toty = v _totsin 110º; thus,

v _wy = (38.0 thou/s)(0.940)−45.0 m/s = −9.29 thousand/s.

This minus sign indicates motion southward which is consistent with the diagram. At present that the perpendicular components of the wind velocity v _westwardx and 5 _wy are known, nosotros can find the magnitude and direction of v_w. First, the magnitude is

[latex]\begin{array}{c}{{v}_{w}}\hfill=\hfill\sqrt{{{five}_{wx}}^{two}+{{five}_{wy}}^{2}} \\\hfill=\hfill\sqrt{({-13.0}\text{ g/s})^{ii} + ({-ix.29}\text{ yard/s}^{two})}\stop{array}[/latex]

so that

5 _w= xvi.0 m/due south.

The direction is:

θ = tan^{− i} (5 _wy /v _w10 ) = tan^{− ane}(−nine.29/−13.0)

giving

θ= 35.6º.

Discussion

The air current's speed and direction are consistent with the significant effect the wind has on the total velocity of the airplane, equally seen in Figure 5. Because the plane is fighting a potent combination of crosswind and head-wind, it ends upward with a total velocity significantly less than its velocity relative to the air mass as well equally heading in a unlike direction.

Note that in both of the last two examples, nosotros were able to make the mathematics easier by choosing a coordinate system with one axis parallel to one of the velocities. We will repeatedly discover that choosing an appropriate coordinate system makes problem solving easier. For instance, in projectile motion nosotros always use a coordinate arrangement with one axis parallel to gravity.

Relative Velocities and Classical Relativity

When calculation velocities, we have been careful to specify that the velocity is relative to some reference frame . These velocities are chosen relative velocities. For example, the velocity of an aeroplane relative to an air mass is different from its velocity relative to the ground. Both are quite unlike from the velocity of an aeroplane relative to its passengers (which should be close to zilch). Relative velocities are one aspect of relativity, which is defined to be the study of how unlike observers moving relative to each other mensurate the aforementioned phenomenon.

Well-nigh everyone has heard of relativity and immediately associates it with Albert Einstein (1879–1955), the greatest physicist of the 20th century. Einstein revolutionized our view of nature with his mod theory of relativity, which we shall study in later chapters. The relative velocities in this section are actually aspects of classical relativity, first discussed correctly by Galileo and Isaac Newton. Classical relativity is limited to situations where speeds are less than about 1% of the speed of light—that is, less than 3,000 km/south. Most things nosotros run into in daily life motion slower than this speed.

Allow us consider an instance of what two different observers see in a situation analyzed long ago by Galileo. Suppose a sailor at the pinnacle of a mast on a moving transport drops his binoculars. Where will it hitting the deck? Volition it hitting at the base of the mast, or will it hit backside the mast because the ship is moving frontwards? The reply is that if air resistance is negligible, the binoculars will hit at the base of the mast at a point directly beneath its point of release. Now let us consider what two different observers see when the binoculars drib. Ane observer is on the send and the other on shore. The binoculars have no horizontal velocity relative to the observer on the ship, and so he sees them fall straight downwards the mast. (See Effigy 6.) To the observer on shore, the binoculars and the ship have the same horizontal velocity, so both motility the aforementioned distance forward while the binoculars are falling. This observer sees the curved path shown in Figure 6. Although the paths await unlike to the different observers, each sees the same result—the binoculars striking at the base of the mast and not behind information technology. To get the correct description, it is crucial to correctly specify the velocities relative to the observer.

Effigy 6. Classical relativity. The same motion equally viewed by two different observers. An observer on the moving ship sees the binoculars dropped from the top of its mast fall straight down. An observer on shore sees the binoculars take the curved path, moving forward with the ship. Both observers meet the binoculars strike the deck at the base of the mast. The initial horizontal velocity is different relative to the two observers. (The ship is shown moving rather fast to emphasize the effect.)

Case iii. Computing Relative Velocity: An Airline Passenger Drops a Coin

An airline passenger drops a coin while the plane is moving at 260 g/south. What is the velocity of the coin when it strikes the floor i.50 one thousand below its point of release: (a) Measured relative to the aeroplane? (b) Measured relative to the Earth?

Figure 7. The motion of a coin dropped inside an airplane equally viewed by two different observers. (a) An observer in the plane sees the money fall straight down. (b) An observer on the ground sees the coin move almost horizontally.

Strategy

Both issues tin can exist solved with the techniques for falling objects and projectiles. In part (a), the initial velocity of the coin is zero relative to the plane, and then the motility is that of a falling object (1-dimensional). In office (b), the initial velocity is 260 chiliad/s horizontal relative to the Earth and gravity is vertical, then this motion is a projectile move. In both parts, it is best to utilize a coordinate arrangement with vertical and horizontal axes.

Solution for (a)

Using the given information, nosotros note that the initial velocity and position are zippo, and the last position is 1.50 one thousand. The final velocity tin can exist institute using the equation:

five _y ^two = 5 _0y ² −2g(y−y ₀).

Substituting known values into the equation, we get

[latex]{{v}_{y}}^{2}={0}^{2}-2\left(ix\text{.}\text{fourscore}{\text{yard/s}}^{2}\right)\left(-ane\text{.}\text{fifty}\text{thou}-0 m\right)=\text{29}\text{.}iv{\text{m}}^{2}{\text{/s}}^{2}[/latex]

yielding

v _y = −5.42 m/southward.

We know that the square root of 29.4 has two roots: 5.42 and -5.42. We choose the negative root because nosotros know that the velocity is directed downwards, and we have defined the positive direction to be up. There is no initial horizontal velocity relative to the plane and no horizontal acceleration, and so the motion is straight downwards relative to the airplane.

Solution for (b)

Considering the initial vertical velocity is null relative to the ground and vertical motility is independent of horizontal motion, the last vertical velocity for the coin relative to the basis is v_y = -five.42 yard/s, the aforementioned as found in role (a). In contrast to function (a), there now is a horizontal component of the velocity. All the same, since in that location is no horizontal acceleration, the initial and final horizontal velocities are the aforementioned and 5 _x =260 m/s. The x– and y-components of velocity tin be combined to find the magnitude of the terminal velocity:

[latex]v=\sqrt{{{v}_{x}}^{2}+{{five}_{y}}^{2}}[/latex].

Thus,

[latex]v=\sqrt{({260\text{ m/southward})}^{2}+({-v.42\text{ m/south}})^{2}}[/latex]

yielding

v= 260.06 thou/s.

The direction is given by:

θ = tan⁻¹(v _y /v _x ) = tan⁻¹(−5.42/260)

then that

θ= tan⁻¹(−0.0208)=−one.19º.

Discussion

In part (a), the final velocity relative to the aeroplane is the aforementioned every bit it would be if the coin were dropped from residuum on the Earth and fell ane.l grand. This outcome fits our experience; objects in a airplane fall the same way when the plane is flying horizontally every bit when it is at rest on the ground. This issue is likewise truthful in moving cars. In part (b), an observer on the ground sees a much different movement for the coin. The plane is moving so fast horizontally to brainstorm with that its concluding velocity is barely greater than the initial velocity. Once over again, we see that in 2 dimensions, vectors practise not add like ordinary numbers—the final velocity v in role (b) is not (260 – 5.42) k/south; rather, information technology is 260.06 m/south. The velocity'due south magnitude had to be calculated to 5 digits to run across any deviation from that of the airplane. The motions as seen past dissimilar observers (one in the plane and one on the ground) in this instance are analogous to those discussed for the binoculars dropped from the mast of a moving ship, except that the velocity of the airplane is much larger, then that the two observers meet very different paths. (See Figure 7.) In improver, both observers run into the coin fall 1.50 thousand vertically, merely the one on the ground also sees information technology move forward 144 m (this calculation is left for the reader). Thus, one observer sees a vertical path, the other a nearly horizontal path.

Making Connections: Relativity and Einstein

Because Einstein was able to clearly ascertain how measurements are made (some involve light) and because the speed of light is the same for all observers, the outcomes are spectacularly unexpected. Time varies with observer, energy is stored equally increased mass, and more than surprises wait.

PhET Explorations: Motility in 2D

Try the new "Ladybug Motion 2D" simulation for the latest updated version. Learn about position, velocity, and acceleration vectors. Motility the brawl with the mouse or allow the simulation move the ball in four types of motion (ii types of linear, simple harmonic, circumvolve).

Click to download. Run using Java.

Summary

• Velocities in two dimensions are added using the same analytical vector techniques, which are rewritten as

  5 _ten = v cosθ

  v _y = v sinθ

  [latex]v=\sqrt{{{v}_{10}}^{2}+{{v}_{y}}^{2}}[/latex]

  θ = tan ^{− ane} (v _y /v _ten ).

• Relative velocity is the velocity of an object as observed from a detail reference frame, and it varies dramatically with reference frame.
• Relativity is the report of how different observers measure the same miracle, particularly when the observers move relative to ane another. Classical relativity is limited to situations where speed is less than well-nigh 1% of the speed of light (3000 km/southward).

Conceptual Questions

1. What frame or frames of reference exercise y'all instinctively use when driving a car? When flying in a commercial jet airplane?
2. A basketball player dribbling down the court ordinarily keeps his eyes fixed on the players around him. He is moving fast. Why doesn't he need to keep his eyes on the brawl?
3. If someone is riding in the dorsum of a pickup truck and throws a softball straight backward, is information technology possible for the ball to fall directly down as viewed past a person standing at the side of the road? Under what status would this occur? How would the motion of the ball announced to the person who threw information technology?
4. The hat of a jogger running at constant velocity falls off the dorsum of his head. Draw a sketch showing the path of the hat in the jogger's frame of reference. Draw its path as viewed past a stationary observer.
5. A clod of dirt falls from the bed of a moving truck. It strikes the basis directly below the end of the truck. What is the direction of its velocity relative to the truck just before information technology hits? Is this the aforementioned as the direction of its velocity relative to ground just earlier it hits? Explain your answers.

Problems & Exercises

1. Bryan Allen pedaled a human-powered aircraft across the English Channel from the cliffs of Dover to Cap Gris-Nez on June 12, 1979. (a) He flew for 169 min at an boilerplate velocity of 3.53 k/s in a direction 45º due south of east. What was his total displacement? (b) Allen encountered a headwind averaging 2.00 m/due south almost precisely in the opposite direction of his motion relative to the World. What was his average velocity relative to the air? (c) What was his total deportation relative to the air mass?

ii. A seagull flies at a velocity of 9.00 m/due south straight into the wind. (a) If it takes the bird xx.0 min to travel 6.00 km relative to the Earth, what is the velocity of the air current? (b) If the bird turns around and flies with the air current, how long volition he have to return 6.00 km? (c) Discuss how the wind affects the total round-trip fourth dimension compared to what information technology would be with no wind.

3. Near the finish of a marathon race, the get-go two runners are separated past a distance of 45.0 thou. The front runner has a velocity of 3.fifty thou/s, and the second a velocity of 4.20 m/southward. (a) What is the velocity of the second runner relative to the first? (b) If the front end runner is 250 m from the finish line, who will win the race, bold they run at constant velocity? (c) What distance ahead will the winner be when she crosses the finish line?

4. Verify that the coin dropped by the airline passenger in Figure 7 travels 144 thou horizontally while falling ane.l grand in the frame of reference of the Earth.

5. A football quarterback is moving directly astern at a speed of 2.00 m/s when he throws a pass to a player 18.0 m direct downfield. The brawl is thrown at an angle of 25.0º relative to the ground and is defenseless at the same height as information technology is released. What is the initial velocity of the ball relative to the quarterback ?

6. A transport sets sheet from Rotterdam, The Netherlands, heading northward at seven.00 thou/s relative to the water. The local ocean current is 1.fifty one thousand/s in a direction 40.0º north of east. What is the velocity of the ship relative to the Earth?

seven. (a) A jet aeroplane flying from Darwin, Australia, has an air speed of 260 chiliad/due south in a direction 5.0º southward of due west. It is in the jet stream, which is blowing at 35.0 k/s in a management 15º south of e. What is the velocity of the airplane relative to the Globe? (b) Discuss whether your answers are consistent with your expectations for the effect of the current of air on the plane's path.

8. (a) In what direction would the send in Exercise 6 take to travel in order to have a velocity direct north relative to the Earth, assuming its speed relative to the water remains 7.00 m/due south? (b) What would its speed exist relative to the Earth?

9. (a) Some other airplane is flying in a jet stream that is blowing at 45.0 m/s in a direction 20º south of e (as in Exercise seven). Its management of motility relative to the Earth is 45º south of due west, while its direction of travel relative to the air is five.00º south of w. What is the plane'southward speed relative to the air mass? (b) What is the airplane's speed relative to the Earth?

10. A sandal is dropped from the tiptop of a 15.0-m-high mast on a ship moving at ane.75 1000/southward due south. Summate the velocity of the sandal when information technology hits the deck of the ship: (a) relative to the ship and (b) relative to a stationary observer on shore. (c) Discuss how the answers give a consistent result for the position at which the sandal hits the deck.

11. The velocity of the wind relative to the water is crucial to sailboats. Suppose a sailboat is in an body of water electric current that has a velocity of 2.20 m/s in a direction 30.0º east of north relative to the Earth. Information technology encounters a wind that has a velocity of four.fifty m/s in a direction of fifty.0º south of west relative to the Earth. What is the velocity of the wind relative to the water?

12. The great astronomer Edwin Hubble discovered that all distant galaxies are receding from our Milky way Galaxy with velocities proportional to their distances. It appears to an observer on the World that we are at the heart of an expanding universe. Figure 9 illustrates this for v galaxies lying along a straight line, with the Milky way Galaxy at the center. Using the data from the effigy, calculate the velocities: (a) relative to galaxy 2 and (b) relative to milky way 5. The results mean that observers on all galaxies volition see themselves at the heart of the expanding universe, and they would probable exist aware of relative velocities, concluding that it is not possible to locate the center of expansion with the given information.

Figure 9. V galaxies on a straight line, showing their distances and velocities relative to the Milky way (MW) Galaxy. The distances are in millions of light years (Mly), where a lite yr is the altitude light travels in one year. The velocities are nearly proportional to the distances. The sizes of the galaxies are profoundly exaggerated; an boilerplate galaxy is about 0.1 Mly beyond.

(a) Employ the distance and velocity data in [link] to find the rate of expansion as a role of distance.

(b) If you extrapolate back in fourth dimension, how long ago would all of the galaxies take been at approximately the same position? The two parts of this problem requite you some idea of how the Hubble constant for universal expansion and the time dorsum to the Big Bang are adamant, respectively.

13. An athlete crosses a 25-1000-wide river by swimming perpendicular to the water electric current at a speed of 0.5 m/s relative to the water. He reaches the opposite side at a distance xl 1000 downstream from his starting point. How fast is the water in the river flowing with respect to the ground? What is the speed of the swimmer with respect to a friend at rest on the ground?

14. A send sailing in the Gulf Stream is heading 25.0º due west of north at a speed of 4.00 g/s relative to the water. Its velocity relative to the Earth is four.80 m/s 5.00º west of north. What is the velocity of the Gulf Stream? (The velocity obtained is typical for the Gulf Stream a few hundred kilometers off the due east coast of the United states of america.)

15. An water ice hockey histrion is moving at eight.00 yard/s when he hits the puck toward the goal. The speed of the puck relative to the actor is 29.0 m/southward. The line between the heart of the goal and the player makes a ninety.0º angle relative to his path as shown in Figure 10. What angle must the puck's velocity make relative to the player (in his frame of reference) to hit the center of the goal?

Effigy 10. An ice hockey role player moving beyond the rink must shoot astern to give the puck a velocity toward the goal.

xvi. Unreasonable Results Suppose you wish to shoot supplies direct upwardly to astronauts in an orbit 36,000 km to a higher place the surface of the World. (a) At what velocity must the supplies be launched? (b) What is unreasonable about this velocity? (c) Is there a problem with the relative velocity between the supplies and the astronauts when the supplies reach their maximum height? (d) Is the premise unreasonable or is the available equation inapplicable? Explain your answer.

17. Unreasonable Results A commercial plane has an air speed of 280 m/s due due east and flies with a strong tailwind. Information technology travels 3000 km in a management 5º due south of due east in 1.fifty h. (a) What was the velocity of the airplane relative to the footing? (b) Calculate the magnitude and management of the tailwind's velocity. (c) What is unreasonable about both of these velocities? (d) Which premise is unreasonable?

18. Construct Your Own Problem Consider an airplane headed for a rail in a cantankerous wind. Construct a problem in which you lot calculate the angle the airplane must wing relative to the air mass in order to accept a velocity parallel to the track. Amongst the things to consider are the management of the runway, the wind speed and management (its velocity) and the speed of the aeroplane relative to the air mass. Too calculate the speed of the airplane relative to the ground. Talk over any last infinitesimal maneuvers the pilot might have to perform in order for the plane to land with its wheels pointing straight downwardly the rail.

Glossary

classical relativity:

the study of relative velocities in situations where speeds are less than about 1% of the speed of low-cal—that is, less than 3000 km/south

relative velocity:

the velocity of an object every bit observed from a particular reference frame

relativity:

the written report of how different observers moving relative to each other measure out the same miracle

velocity:

speed in a given direction

vector addition:

the rules that apply to calculation vectors together

Selected Solutions to Issues & Exercises

ane. (a) 35.8 km, 45º south of east (b) 5.53 m/s, 45º south of eastward (c) 56.1 km, 45º south of east

three. (a) 0.70 m/s faster (b) Second runner wins (c) iv.17 g

5. 17.0 yard/due south, 22.1º

seven.  (a) 230 k/s, 8.0º s of west (b) The current of air should make the plane travel slower and more to the south, which is what was calculated.

9. (a) 63.five m/s (b) 29.six m/due south

11. 6.68 m/due south, 53.3º due south of west

12.  (a) [latex]{H}_{\text{boilerplate}}=\text{xiv}\text{.}\text{9}\frac{\text{km/due south}}{\text{Mly}}[/latex] (b) 20.2 billion years

fourteen. 1.72 yard/south, 42.3º north of e

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