In this worksheet, we will practice expressing velocities as vector components, combining and decomposing vectors to find resultants in particular directions.

**Q1: **

An athlete crosses a 30.0 m wide river by swimming perpendicular to the water current at a speed of 0.500 m/s relative to the water. He reaches the opposite side at a distance 75.0 m 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 bank of the river?

**Q2: **

The position of a particle for is given by

What is the particleβs velocity at ?

- A m/s
- B m/s
- C m/s
- D m/s
- E m/s

What is the particleβs speed at ?

What is the particleβs speed at ?

What is the average velocity of the particle between and ?

- A m/s
- B m/s
- C m/s
- D m/s
- E m/s

**Q3: **

A boat leaves a dock at the time , heading out into a lake. The boat accelerates at . A strong wind pushes the boat, giving it an additional velocity m/s.

What is the net velocity of the boat at s?

- A m/s
- B m/s
- C m/s
- D m/s
- E m/s

What is the displacement relative to the dock of the boat at s?

- A m
- B m
- C m
- D m
- E m

**Q5: **

A droneβs average velocity vector . What is the droneβs average speed?

**Q6: **

A Lockheed Martin F-35 II Lightning jet takes off from an aircraft carrier with a runway length of 95 m. The jetβs takeoff speed at the end of the runway is m/s. At the point of leaving the deck of the aircraft carrier, the F-35βs acceleration decreases to a constant acceleration of 4.5 m/s^{2} at above the horizontal.

What is the initial acceleration of the F-35 on the deck of the aircraft carrier to make it airborne?

At what altitude is the fighter 7.3 s after it leaves the deck of the aircraft carrier?

What is the velocity of the fighter 7.3 s after it leaves the deck of the aircraft carrier?

- A m/s
- B m/s
- C m/s
- D m/s
- E m/s

How far has the fighter traveled horizontally 7.3 s after it leaves the deck of the aircraft carrier?

**Q7: **

A bird takes off and is carried by a strong wind. The bird travels straight northeast a distance of 95.0 km for 2.17 h before landing. Consider the -axis as due east and the -axis as due north.

What is the displacement from where the birdβs flight begins to where the it ends?

- A km
- B km
- C km
- D km
- E km

What is the average velocity of the bird?

- A km/h
- B km/h
- C km/h
- D km/h
- E km/h

**Q8: **

A small plane can fly at 175 km/h in still air. The plane flies in a wind that blows directly out of the west at 36 km/h.

At what angle west of north must the plane point in order for it to move directly north?

How much time is needed for the plane to reach a point 300 km directly north of its current position?

**Q9: **

A ship sailing in the Gulf Stream is heading west of north at a speed of 4.00 m/s relative to the water. Its velocity relative to Earth is 4.80 m/s, 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 east coast of the United States.)

- A 1.43 m/s, east of north
- B 1.27 m/s, north of east
- C 1.98 m/s, east of north
- D 1.72 m/s, north of east
- E 1.10 m/s, north of east

**Q10: **

The velocity of the wind relative to the water is crucial to sailing boats. Suppose a sailing boat is in an ocean current that has a velocity of 2.20 m/s in a direction east of north relative to Earth. It encounters a wind that has a velocity of 4.50 m/s in a direction of south of west relative to Earth.

What is the speed of the wind relative to the water?

- A 6.12 m/s
- B 5.69 m/s
- C 8.94 m/s
- D 6.68 m/s
- E 4.56 m/s

What is the direction of the wind, south of west, relative to the water?

- A
- B
- C
- D
- E

**Q11: **

A sandal is dropped from the top of an 18.0 m high mast on a ship moving at 3.05 m/s due south.

Calculate the velocity of the sandal relative to the ship when it hits the deck of the ship.

Calculate the velocity of the sandal relative to a stationary observer on the shore.

**Q12: **

A convoy of vehicles has a velocity vector . What is the unit vector of the convoyβs direction of motion?

- A
- B
- C
- D
- E

**Q13: **

Two motorboats piloted by Isabella and Jacob were moving on a lake. Their motion was recorded at different times. At 9:00, Isabella was moving north at 6 knots and Jacob was moving west at 6 knots. At 9:30, Isabella was moving west at 6 knots and Jacob was moving west at 3 knots. At 10:00, Isabella was moving northeast at 6 knots and Jacob was moving south at 3 knots. At 10:30, Isabella was moving northeast at 6 knots and Jacob was moving southwest at 6 knots. Finally, at 11:00, both were moving northeast at 2 knots but Jacob was moving closer to the shore.

At 9:00, were Isabella and Jacobβs velocity vectors equal to or unequal to each other?

- Aunequal
- Bequal

At 9:30, were Isabella and Jacobβs velocity vectors equal to or unequal to each other?

- Aunequal
- Bequal

At 10:00, were Isabella and Jacobβs velocity vectors equal to or unequal to each other?

- Aequal
- Bunequal

At 10:30, were Isabella and Jacobβs velocity vectors equal to or unequal to each other?

- Aequal
- Bunequal

At 11:00, were Isabella and Jacobβs velocity vectors equal to or unequal to each other?

- Aequal
- Bunequal

**Q14: **

An aeroplane is flying through a jet stream that is blowing at 45.0 m/s in a direction south of east. The direction of the planeβs motion relative to Earth is south of west, while its direction of travel relative to the air is south of west.

What is the aeroplaneβs speed relative to the air mass?

- A 57.4 m/s
- B 61.2 m/s
- C 71.8 m/s
- D 63.5 m/s
- E 69.4 m/s

What is the aeroplaneβs speed relative to Earth?

- A 29.6 m/s
- B 31.4 m/s
- C 19.8 m/s
- D 25.6 m/s
- E 27.2 m/s

**Q15: **

A ship sets sail from Portsmouth, England. The local ocean current is 2.50 m/s in a direction south of west. If the ship is travelling at 8.00 m/s relative to the water, in what direction, east of south, would it have to travel in order to have a velocity straight south relative to Earth?

- A
- B
- C
- D
- E

**Q16: **

A jet aeroplane flying from Darwin, Australia, has an airspeed of 260 m/s in a direction south of west. It is in the jet stream, which is blowing at 35.0 m/s in a direction south of east.

What is the speed of the aeroplane relative to Earth?

- A 180 m/s
- B 260 m/s
- C 290 m/s
- D 230 m/s
- E 250 m/s

In what direction is the aeroplane travelling, south of west, relative to Earth?

- A
- B
- C
- D
- E

**Q17: **

A ship sets sail from Rotterdam, The Netherlands, heading due north at 8.00 m/s relative to the water. The local ocean current is 2.50 m/s in a direction north of east. What is the speed of the ship relative to Earth?

**Q18: **

A seagull flies at a velocity of 8.00 m/s straight into the wind. It takes the bird 24.0 min to travel 5.00 km relative to Earth.

What is the speed of the wind?

If the bird turns around and flies with the wind, how long, in minutes, will it take to return 5.00 km?

**Q19: **

Bryan Allen pedaled a human-powered aircraft across the English Channel from the cliffs of Dover to Cap Gris-Nez on June 12, 1,979. He flew for 182 min at an average velocity of 3.17 m/s relative to Earth in a direction south of east.

What was the magnitude of his displacement in kilometers?

Allen encountered a headwind averaging 2.50 m/s almost precisely in the opposite direction of his motion relative to Earth. What was his average speed relative to the air?

What was the magnitude of his displacement relative to the air mass in kilometers?

- A 48.9 km
- B 34.6 km
- C 61.9 km
- D 27.3 km
- E 55.4 km

**Q20: **

In the diagram, the two velocities and add to give the total velocity . For each part of the question, give the velocities in units of meters per second.

Find the magnitude of .

Find the magnitude of .

What is the -component of ?

What is the -component of ?

Consider that is rotated clockwise about the origin. By how much does the magnitude of change?