## Mastering Physics Solutions Chapter 3 Vectors In Physics

**Chapter 3 Vectors In Physics Q.1CQ**

For the following quantities, indicate which is a scalar and which is a vector: (a) the time it takes for you to run the 100-yard dash; (b) your displacement after running the 100-yard dash; (c) your average velocity while running; (d) your average speed while running.

**Solution:**

(A) Scalar, since there is no direction.

(B) Vector, since displacement depends on the direction.

(C) Vector, since velocity is a vector quantity.

(D) Scalar, since speed is a scalar quantity.

**Chapter 3 Vectors In Physics Q.1P**

Suppose that the component of a certain vector is doubled, (a) By what multiplicative factor docs the magnitude of the vector change? (b) By what multiplicative factor does the direction angle of the vector change?

**Solution:**

**Chapter 3 Vectors In Physics Q.2CQ**

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**Chapter 3 Vectors In Physics Q.2P**

**Solution:**

CONCEPT:

The physical quantity which has both magnitude and direction is called as vector quantity. It is represented by an arrow such that the length of the arrow is proportional to the magnitude of the physical quantity and the arrow head is pointed in the given direction.

**Chapter 3 Vectors In Physics Q.3CQ**

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**Chapter 3 Vectors In Physics Q.3P**

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**Chapter 3 Vectors In Physics Q.4CQ**

Can a component of a vector be greater than the vector’s magnitude?

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**Chapter 3 Vectors In Physics Q.4P**

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**Chapter 3 Vectors In Physics Q.5CQ**

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**Chapter 3 Vectors In Physics Q.5P**

The press box at a baseball park is 32.0 ft above the ground. A reporter in the press box looks at an angle of 15.0° below the horizontal to see second base. What is the horizontal distance from the press box to second base?

**Solution:**

**Chapter 3 Vectors In Physics Q.6CQ**

Can a vector with zero magnitude have one or more components that are nonzero? Explain.

**Solution:**

No, if a vector has a non-zero component, the smallest magnitude it can have is the magnitude of the component.

**Chapter 3 Vectors In Physics Q.6P**

You are driving up a long, inclined road. After 1.2 miles you notice that signs along the roadside indicate that your elevation has increased by 530 ft. (a) What is the angle of the road above the horizontal? (b) How far do you have to drive to gam an additional 150 ft of elevation?

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**Chapter 3 Vectors In Physics Q.7CQ**

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**Chapter 3 Vectors In Physics Q.7P**

A One-Percent Grade A road that rises 1 ft for every 100 ft traveled horizontally is said to have a 1 % grade. Portions of the Lewiston grade, near Lewiston, Idaho, have a 6% grade. At what angle is this road inclined above the horizontal?

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**Chapter 3 Vectors In Physics Q.8CQ**

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**Chapter 3 Vectors In Physics Q.8P**

Find the x and y components of a position vector of magnitude r = 75 m, if its angle relative to the x axis is (a) 35.0° and (b) 65.0°.

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**Chapter 3 Vectors In Physics Q.9CQ**

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**Chapter 3 Vectors In Physics Q.9P**

A baseball “diamond” (Figure) is a square with sides 90 ft in length. If the positive x axis points from home plate to first base, and the positive y axis points from home plate to third base, find the displacement vector of a base runner who has just hit (a) a double, (b) a triple, or (c) a home run.

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**Chapter 3 Vectors In Physics Q.10CQ**

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**Chapter 3 Vectors In Physics Q.10P**

A lighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base, as shown in Figure. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30.0° above the horizontal. If the sailor’s eye level is 14 ft above the water, how far is the ship from the rocks?

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**Chapter 3 Vectors In Physics Q.11CQ**

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**Chapter 3 Vectors In Physics Q.11P**

H2 O A water molecule is shown schematically in Figure. The distance from the center of the oxygen atom to the center of a hydrogen atom is 0.96 Å, and the angle between the hydrogen atoms is 1.04.5°. Find the center-to-center distance between the hydrogen atoms. (1 Å = 10-10 m.)

**Solution:**

**Chapter 3 Vectors In Physics Q.12CQ**

Use a sketch to show that two vectors of unequal magnitude cannot add to zero, but that three vectors of unequal magnitude can.

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**Chapter 3 Vectors In Physics Q.12**

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**Chapter 3 Vectors In Physics Q.13CQ**

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**Chapter 3 Vectors In Physics Q.13P**

Rain is failing vertically downward and you are running for shelter. To keep driest, should you hold your umbrella vertically, tilted forward, or tilted backward? Explain.

**Solution:**

To keep the driest, tilt the umbrella forward and let it point in the direction opposite to the direction of the rain’s velocity in reference to you.

**Chapter 3 Vectors In Physics Q.14CQ**

When sailing, the wind feels stronger when you sail upwind (“beating”) than when you are sailing downwind (“running”). Explain.

**Solution:**

**Chapter 3 Vectors In Physics Q.14P**

You drive a car 680 ft to the east, then 340 ft to the north. (a) What is the magnitude of your displacement? (b) Using a sketch, estimate the direction of your displacement, (c) Verify your estimate in part (b) with a numerical calculation of the direction.

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**Chapter 3 Vectors In Physics Q.15P**

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**Chapter 3 Vectors In Physics Q.16P**

A treasure map directs you to start at a palm tree and walk due north for 15.0 m. You are then to turn 90° and walk 22.0 m; then turn 90° again and walk 5.00 m. Give the distance from the palm tree, and the direction relative to north, for the of the four possible locations of the treasure.

**Solution:**

**Chapter 3 Vectors In Physics Q.17P**

A whale comes to the surface to breathe and then dives at an angle of 20.0° below the horizontal (Figure). If the whale continues in a straight line for 150 m, (a) how deep is it, and (b) how far has it traveled horizontally?

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**Chapter 3 Vectors In Physics Q.18P**

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**Chapter 3 Vectors In Physics Q.19P**

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**Chapter 3 Vectors In Physics Q.20P**

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**Chapter 3 Vectors In Physics Q.21P**

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**Chapter 3 Vectors In Physics Q.22P**

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**Chapter 3 Vectors In Physics Q.23P**

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**Chapter 3 Vectors In Physics Q.24P**

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**Chapter 3 Vectors In Physics Q.25P**

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**Chapter 3 Vectors In Physics Q.26P**

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**Chapter 3 Vectors In Physics Q.27P**

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**Chapter 3 Vectors In Physics Q.28P**

A vector has a magnitude of 3.50 m and points in a direction that is 145° counterclockwise from the x axis. Find the x and y components of this vector.

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**Chapter 3 Vectors In Physics Q.29P**

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**Chapter 3 Vectors In Physics Q.30P**

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**Chapter 3 Vectors In Physics Q.31P**

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**Chapter 3 Vectors In Physics Q.32P**

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**Chapter 3 Vectors In Physics Q.33P**

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**Chapter 3 Vectors In Physics Q.34P**

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**Chapter 3 Vectors In Physics Q.35P**

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**Chapter 3 Vectors In Physics Q.36P**

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**Chapter 3 Vectors In Physics Q.37P**

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**Chapter 3 Vectors In Physics Q.38P**

In its daily prowl of the neighborhood, a cat makes a displacement of 120 m due north, followed by a 72-m displacement due west, (a) Find the magnitude and direction of the displacement required for the cat to return home, (b) If, instead, the cat had first prowled 72 m west and then 120 m north, how would this affect the displacement needed to bring it home? Explain.

**Solution:**

**Chapter 3 Vectors In Physics Q.39P**

If the cat in Problem takes 45 minutes to complete the 120-m displacement and 17 minutes to complete the 72-m displacement, what arc the magnitude and direction of its average velocity during this 62-minute period of time?

In its daily prowl of the neighborhood, a cat makes a displacement of 120 m due north, followed by a 72-m displacement due west, (a) Find the magnitude and direction of the displacement required for the cat to return home, (b) If, instead, the cat had first prowled 72 m west and then 120 m north, how would this affect the displacement needed to bring it home? Explain.

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**Chapter 3 Vectors In Physics Q.40P**

What are the direction and magnitude of your total displacement if you have traveled due west with a speed of 27 m/s for 125 s, their due south at 14 m/s for 66 s?

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**Chapter 3 Vectors In Physics Q.41P**

You drive a car 1500 ft to the east, then 2500 ft to the north. If the trip took 3.0 minutes, what were the direction and magnitude of your average velocity?

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**Chapter 3 Vectors In Physics Q.42P**

A jogger runs with a speed of 3.25 m/s in a direction 30.0° above the x axis, (a) Find the x and y components of the jogger’s velocity, (b) How will the velocity components found in part (a) change if the jogger’s speed is halved?

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**Chapter 3 Vectors In Physics Q.43P**

You throw a ball upward with an initial speed of 4.5 m/s. When it returns to your hand 0.92 s later, it has the same speed in the downward direction (assuming air resistance can be ignored). What was the average acceleration vector of the ball?

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**Chapter 3 Vectors In Physics Q.44P**

A skateboarder rolls from rest down an inclined ramp that is 15.0 m long and inclined above the horizontal at an angle of θ = 20.0°. When she reaches the bottom of the ramp 3.00 s later her speed is 10.0 m/s. Show that the average acceleration of the skateboarder is g sin θ, where g = 9.81 m/s2.

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**Chapter 3 Vectors In Physics Q.45P**

Consider a skateboarder who starts from rest at the top of a ramp that is inclined at an angle of 17.5° to the horizontal. Assuming that the skateboarder’s acceleration is g sin 17.5°, Find his speed when he reaches the bottom of the ramp in 3.25 s.

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**Chapter 3 Vectors In Physics Q.46P**

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**Chapter 3 Vectors In Physics Q.47P**

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**Chapter 3 Vectors In Physics Q.48P**

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**Chapter 3 Vectors In Physics Q.49P**

As an airplane taxies on the runway with a speed of 16.5 m/s, a Flight attendant walks toward the tail of the plane with a speed of 1.22 m/s. What is the flight attendant’s speed relative to the ground?

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**Chapter 3 Vectors In Physics Q.50P**

Referring to part (a) of Example, find the time it takes for the boat to reach the opposite shore if the river is 35 m wide.

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**Chapter 3 Vectors In Physics Q.51P**

As you hurry to catch your flight at the local airport, you encounter a moving walkway that is 85 m long and has a speed of 2.2 m/s relative to the ground. If it takes you 68 s to cover 85 m when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.

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**Chapter 3 Vectors In Physics Q.52P**

In Problem, how long would it take you to cover the 85-m length of the walkway if, once you get on the walkway, you immediately turn around and start walking in the opposite direction with a speed of 1.3 m/s relative to the walkway?

As you hurry to catch your flight at the local airport, you encounter a moving walkway that is 85 m long and has a speed of 2.2 m/s relative to the ground. If it takes you 68 s to cover 85 m when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.

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**Chapter 3 Vectors In Physics Q.53P**

The pilot of an airplane wishes to fly due north, but there is a 65-km/h wind blowing toward the east, (a) In what direction should the pilot head her plane if its speed relative to the air is 340 km/h? (b) Draw a vector diagram that illustrates your result in part (a), (c) If the pilot decreases the air speed of the plane, but still wants to head due north, should the angle found in part (a) be increased or decreased?

**Solution:**

**Chapter 3 Vectors In Physics Q.54P**

A passenger walks from one side of a ferry to the other as it approaches a dock. If the passenger’s velocity is 1.50 m/s due north relative to the ferry, and 4.50 m/s at an angle of 30.0° west of north relative to the water, what are the direction and magnitude of the Ferry’s velocity relative to the water?

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**Chapter 3 Vectors In Physics Q.55P**

You are riding on a Jet Ski at an angle of 35° upstream, on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown in Example.)

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**Chapter 3 Vectors In Physics Q.56P**

In Problem, suppose the Jet Ski is moving at a speed of 12 m/s relative to the water, (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis shown, in Example.)

You are riding on a Jet Ski at an angle of 35° upstream, on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown in Example.)

**Solution:**

**Chapter 3 Vectors In Physics Q.57P**

Two people take identical Jet Skis across a river, traveling at the same speed relative to the water. Jet Ski A heads directly across the river and is carried downstream by the current before teaching the opposite shore. Jet Ski B travels in a direction that is 35° upstream and arrives at the opposite shore directly across from the starting point, (a) Which Jet Ski reaches the opposite shore in the least amount of time? (b) Confirm your answer to part (a) by finding the ratio of the time it takes for the two Jet Skis to cross the river. (Note: Angles are measured relative to the x axis shown in Example.)

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**Chapter 3 Vectors In Physics Q.58GP**

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**Chapter 3 Vectors In Physics Q.59GP**

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**Chapter 3 Vectors In Physics Q.60GP**

You slide a box up a loading ramp that is 10.0 ft long. At the top of the ramp the box has risen a height of 3.00 ft. What is the angle of the ramp above the horizontal?

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**Chapter 3 Vectors In Physics Q.61GP**

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**Chapter 3 Vectors In Physics Q.62GP**

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**Chapter 3 Vectors In Physics Q.63GP**

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**Chapter 3 Vectors In Physics Q.64GP**

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**Chapter 3 Vectors In Physics Q.65GP**

Two students perform an experiment with a train and a ball. Michelle tides on a flatcar pulled at 8.35 m/s by a train on a straight, horizontal track; Gary stands at rest on the ground near the tracks. When Michelle throws the ball with an initial angle of 65.0° above the horizontal, from her point of view, Gary sees the ball rise straight up and back down above a fixed point on the ground, (a) Did Michelle throw the ball toward the front of the train or toward the rear of the train? Explain, (b) What was the initial speed of Michelle’s throw? (c) What was the initial speed of the ball as seen by Gary?

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**Chapter 3 Vectors In Physics Q.66GP**

An off-roader explores the open desert in her Hummer. First she drives 25° west of north with a speed of 6.5 km/h for 15 minutes, then due east with a speed of 12 km/h for 7.5 minutes. She completes the final leg of her trip in 22 minutes. What are the direction and speed of travel on the final leg? (Assume her speed is constant on the leg, and that she returns to her starting point at the end of the final leg.)

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**Chapter 3 Vectors In Physics Q.67GP**

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**Chapter 3 Vectors In Physics Q.68GP**

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**Chapter 3 Vectors In Physics Q.69GP**

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**Chapter 3 Vectors In Physics Q.70GP**

Two airplanes taxi as they approach the terminal. Plane 1 taxies with a speed of 12 m/s due north. Plane 2 taxies with a speed of 7.5 m/s in a direction 20° north of west, (a) What are the direction and magnitude of the velocity of plane 1 relative to plane 2? (b) What are the direction and magnitude of the velocity of plane 2 relative to plane 1?

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**Chapter 3 Vectors In Physics Q.71GP**

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**Chapter 3 Vectors In Physics Q.72GP**

Initially, a particle is moving at 4.10 m/s at an angle of 33.5° above the horizontal. Two seconds later, its velocity is 6.05 m/s at an angle of 59.0° below the horizontal. What was the particle’s average acceleration during these 2.00 seconds?

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**Chapter 3 Vectors In Physics Q.73GP**

A passenger on a stopped bus notices that rain is falling vertically just outside the window. When the bus moves with constant velocity, the passenger observes that the falling raindrops are now making an angle of 15° with respect to the vertical. (a) What is the ratio of the speed of the raindrops to the speed of the bus? (b) Find the speed of the raindrops, given that the bus is moving with a speed of 18m/s.

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**Chapter 3 Vectors In Physics Q.74GP**

A Big Clock The clock that rings the bell known as Big Ben has an hour hand that is 9.0 feet long and a minute hand that is 14 feet long, where the distance is measured from the center of the clock to the tip of the hand. What is the tip-to-tip distance between these two hands when the clock reads 12 minutes after four o’clock?

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**Chapter 3 Vectors In Physics Q.75GP**

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**Chapter 3 Vectors In Physics Q.76GP**

Referring to Example, (a) what heading must the boat have if it is to land directly across the river from its starting point? (b) How much time is required for this trip if the river is 25.0 m wide? (c) Suppose the speed of the boat is increased, but it is still desired to land directly across from the starting point. Should the boat’s heading be more upstream, more downstream, or the saine as in part (a)? Explain.

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**Chapter 3 Vectors In Physics Q.77GP**

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**Chapter 3 Vectors In Physics Q.78GP**

As two boats approach the marina, the velocity of boat 1 relative to boat 2 is 2.15 m/s in a direction 47.0° east of north. If boat 1 has a velocity that is 0.775 m/s due north, what is the velocity (magnitude and direction) of boat 2?

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**Chapter 3 Vectors In Physics Q.79PP**

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**Chapter 3 Vectors In Physics Q.80PP**

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**Chapter 3 Vectors In Physics Q.81PP**

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**Chapter 3 Vectors In Physics Q.82IP**

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**Chapter 3 Vectors In Physics Q.83IP**

Referring to Example Suppose the speed of the boat relative to the water is 7.0 m/s. (a) At what angle to the x axis must the boat be headed if it is to land directly across the river from its starting position? (b) If the speed of the boat relative to the water is increased, will the angle needed to go directly across the river increase, decrease, or stay the same? Explain.

**Solution:**

**Chapter 3 Vectors In Physics Q.84IP**

Referring to Example Suppose the boat has a speed of 6.7 m/s relative to the water, and that the dock on the opposite shore of the river is at the location x = 55 m and y = 28 m relative to the starting point of the boat, (a) At what angle relative to the x axis must the boat be pointed in order to reach the other dock? (b) With the angle found in part (a), what is the speed of the boat relative to the ground?

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