Q: A plane has an eastward heading at a speed of 160 m/s relative to the air. If a wind is blowing southward at 20 m/s, what is the velocity of the plane relative to the ground?

$$157\ \text{m/s} at an $$angle $$7.31^\circ$$ south of east

Q: A small boat is moving at a velocity of $$3.0\ \text{m/s}$$ when it is accelerated by a river current perpendicular to its initial direction at $$0.750\ \text{m/s}^2. $$What will be the new velocity of the boat after $$30\ \text{s}$$?

$$25.3\ \text{m/s} at$$ $$82.4^\circ$$ from initial direction

Q: A car travels east at $$20\ \text{m/s}$$ and a truck travels north at $$15\ \text{m/s}. $$What is the velocity of the car relative to the truck?

$$25\ \text{m/s} at$$ $$36.9^\circ$$ east of north

Q: A cyclist moves at $$10\ \text{m/s}$$ north. A wind blows at $$5\ \text{m/s}$$ west. What is the cyclist's velocity relative to the ground?

$$11.2\ \text{m/s} at$$ $$26.6^\circ$$ west of north

Q: A river flows east at $$2\ \text{m/s}. A $$swimmer heads north at $$3\ \text{m/s}$$ relative to the water. What is the swimmer's velocity relative to the ground?

$$3.6\ \text{m/s} at$$ $$33.7^\circ$$ east of north

Q: A plane flies at $$200\ \text{m/s}$$ north relative to the air. If a wind blows at $$50\ \text{m/s}$$ east, what is the plane's velocity relative to the ground?

$$206\ \text{m/s} at$$ $$14.0^\circ$$ east of north

Q: A cyclist moves at $$12\ \text{m/s}$$ west. A wind blows at $$5\ \text{m/s}$$ south. What is the cyclist's velocity relative to the ground?

$$13\ \text{m/s} at$$ $$22.6^\circ$$ south of west

Q: A boat heads north at $$4\ \text{m/s}$$ relative to the water. The river flows east at $$3\ \text{m/s}. $$What is the boat's velocity relative to the ground?

$$5\ \text{m/s} at$$ $$36.9^\circ$$ east of north

Question 1

An airplane needs to fly between two cities that are 500 km apart, with the destination directly north of the origin. If a constant wind blows at 10.0 m/s due west and the plane's airspeed is 50.0 m/s, what direction should the plane head relative to north to reach its destination?

Accepted Answer

4.78° east of north

Question 2

A plane has an eastward heading at a speed of 160 m/s relative to the air. If a wind is blowing southward at 20 m/s, what is the velocity of the plane relative to the ground?

Accepted Answer

$$157\ 	ext{m/s} at an $$angle $$7.31^\circ$$ south of east

Question 3

A small boat is moving at a velocity of $$3.0\ 	ext{m/s}$$ when it is accelerated by a river current perpendicular to its initial direction at $$0.750\ 	ext{m/s}^2. $$What will be the new velocity of the boat after $$30\ 	ext{s}$$?

Accepted Answer

$$25.3\ 	ext{m/s} at$$ $$82.4^\circ$$ from initial direction

Question 4

A ball is tied to the end of a cable and spun in a circle with a radius of $$0.85\ 	ext{m}$$, making $$7.00$$ revolutions every $$5.00\ 	ext{s}. $$What is the magnitude of the acceleration of the ball?

Accepted Answer

$$38.7\ 	ext{m/s}^2$$

Question 5

A satellite orbits the Earth at a distance of $$1.50 	imes 10^7$\ 	ext{m}$$ above the surface. If the Earth's radius is $$6.37 	imes 10^6$\ 	ext{m}$$ and the satellite takes $$6.00 	imes 10^3$\ 	ext{s}$$ for each revolution, what is the acceleration of the satellite?

Accepted Answer

$$0.870\ 	ext{m/s}^2$$

Question 6

A disk-shaped space station $$175\ 	ext{m} in $$diameter spins about its center. How many revolutions per minute (rpm) must it make so that the acceleration at the rim is $$\frac{g}{2}$$?

Accepted Answer

$$2.26\ 	ext{rpm}$$

Question 7

A car travels east at $$20\ 	ext{m/s}$$ and a truck travels north at $$15\ 	ext{m/s}. $$What is the velocity of the car relative to the truck?

Accepted Answer

$$25\ 	ext{m/s} at$$ $$36.9^\circ$$ east of north

Question 8

A cyclist moves at $$10\ 	ext{m/s}$$ north. A wind blows at $$5\ 	ext{m/s}$$ west. What is the cyclist's velocity relative to the ground?

Accepted Answer

$$11.2\ 	ext{m/s} at$$ $$26.6^\circ$$ west of north

Question 9

A ball is swung in a horizontal circle of radius $$0.5\ 	ext{m} at a $$constant speed of $$4\ 	ext{m/s}. $$What is the centripetal acceleration?

Accepted Answer

$$32\ 	ext{m/s}^2$$

Question 10

A river flows east at $$2\ 	ext{m/s}. A $$swimmer heads north at $$3\ 	ext{m/s}$$ relative to the water. What is the swimmer's velocity relative to the ground?

Accepted Answer

$$3.6\ 	ext{m/s} at$$ $$33.7^\circ$$ east of north

Question 11

A train moves at $$30\ 	ext{m/s}$$ east. A passenger walks at $$2\ 	ext{m/s}$$ west inside the train. What is the passenger's velocity relative to the ground?

Accepted Answer

$$28\ 	ext{m/s}$$ east

Question 12

A satellite orbits at a radius $$r$$ from the center of the Earth. Which equation gives the centripetal acceleration $$a_c in $$terms of orbital speed $$v$$?

Accepted Answer

$$a_c = \frac{v^2$}{r}$$

Question 13

A ball is spun in a circle of radius $$0.6\ 	ext{m} at$$ $$5\ 	ext{m/s}. $$What is the tension in the string if the ball has mass $$0.2\ 	ext{kg}$$?

Accepted Answer

$$8.33\ 	ext{N}$$

Question 14

A space station is designed so that the artificial gravity at its rim equals $$g/2. If $$the radius is $$87.5\ 	ext{m}$$, what is the required angular velocity $$\omega in $$rad/s?

Accepted Answer

$$0.75\ 	ext{rad/s}$$

Question 15

A plane flies at $$200\ 	ext{m/s}$$ north relative to the air. If a wind blows at $$50\ 	ext{m/s}$$ east, what is the plane's velocity relative to the ground?

Accepted Answer

$$206\ 	ext{m/s} at$$ $$14.0^\circ$$ east of north

Question 16

A cyclist moves at $$12\ 	ext{m/s}$$ west. A wind blows at $$5\ 	ext{m/s}$$ south. What is the cyclist's velocity relative to the ground?

Accepted Answer

$$13\ 	ext{m/s} at$$ $$22.6^\circ$$ south of west

Question 17

A ball is spun in a circle of radius $$0.4\ 	ext{m} at$$ $$6\ 	ext{m/s}. $$What is the period of revolution?

Accepted Answer

$$0.42\ 	ext{s}$$

Question 18

A boat heads north at $$4\ 	ext{m/s}$$ relative to the water. The river flows east at $$3\ 	ext{m/s}. $$What is the boat's velocity relative to the ground?

Accepted Answer

$$5\ 	ext{m/s} at$$ $$36.9^\circ$$ east of north

Question 19

A satellite orbits the Earth at a radius $$r$$ and period $$T. $$Which equation gives its orbital speed $$v$$?

Accepted Answer

$$v = \frac{2\pi r}{T}$$

Step-by-Step Guidance for Circular Motion and Relative Velocity Problems

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