Q: If $$\vec{v}_{PG} = \vec{v}_{PT} + \vec{v}_{TG}$$, and $$\vec{v}_{PT} = 0.24\ \mathrm{m/s}\ \hat{y}$$, $$\vec{v}_{TG} = 0.68\ \mathrm{m/s}\ \hat{x}$$, what is the angle $$\theta of$$ $$\vec{v}_{PG}$$ with respect to the $$x-$$axis?

$$\theta = \tan$$^{-1}$$\left(\frac{0.24}{0.68}\right) = 19^\circ$$

Question 1

A person walks inside a train at $$1.2\ \mathrm{m/s}$$ relative to the train. The train moves at $$15.0\ \mathrm{m/s}$$ relative to the ground. If the person walks in the same direction as the train's motion, what is the person's velocity relative to the ground?

Accepted Answer

$$16.2\ \mathrm{m/s}$$

Question 2

A person walks inside a train at $$1.2\ \mathrm{m/s}$$ relative to the train, but in the opposite direction to the train's motion. The train moves at $$15.0\ \mathrm{m/s}$$ relative to the ground. What is the person's velocity relative to the ground?

Accepted Answer

$$13.8\ \mathrm{m/s}$$

Question 3

Which equation correctly expresses the velocity of object 1 relative to object 3 in terms of other relative velocities?

Accepted Answer

$$\vec{v}_{13} = \vec{v}_{12} + \vec{v}_{23}$$

Question 4

If $$\vec{v}_{ab} is $$the velocity of object $$a$$ relative to $$b$$, what is the velocity of object $$b$$ relative to $$a$$?

Accepted Answer

$$\vec{v}_{ba} = -\vec{v}_{ab}$$

Question 5

A person climbs a vertical ladder at $$0.24\ \mathrm{m/s}$$ relative to the ladder. The ladder is attached to a train moving at $$0.68\ \mathrm{m/s} in $$the $$+x$$ direction. What is the magnitude of the person's velocity relative to the ground?

Accepted Answer

$$0.72\ \mathrm{m/s}$$

Question 6

A boat moves with a velocity of $$6.1\ \mathrm{m/s}$$ relative to the water, pointing $$25^\circ$$ upstream. The river flows at $$1.4\ \mathrm{m/s}$$ downstream. What is the $$y-$$component of the boat's velocity relative to the ground?

Accepted Answer

$$1.2\ \mathrm{m/s}$$

Question 7

Given $$\vec{v}_{BW} = (6.1\ \mathrm{m/s})[\cos 25^\circ \hat{x} + \sin 25^\circ \hat{y}]$$ and $$\vec{v}_{WG} = -1.4\ \mathrm{m/s}\ \hat{y}$$, what is $$\vec{v}_{BG}$$?

Accepted Answer

$$(5.5\ \mathrm{m/s})\hat{x} + (1.2\ \mathrm{m/s})\hat{y}$$

Question 8

A boat wants to move straight across a river (perpendicular to the shore) with a river current of $$1.4\ \mathrm{m/s}. If $$the boat's speed relative to the water is $$6.1\ \mathrm{m/s}$$, at what angle upstream should the boat point?

Accepted Answer

$$13^\circ$$

Question 9

A boat moves directly across a river (in the $$+x$$ direction) with a velocity of $$5.0\ \mathrm{m/s}$$ relative to the ground. The river flows at $$1.4\ \mathrm{m/s} in $$the $$-y$$ direction. What is the magnitude of the boat's velocity relative to the water?

Accepted Answer

$$5.2\ \mathrm{m/s}$$

Question 10

A boat moves directly across a river (in the $$+x$$ direction) with a velocity of $$5.0\ \mathrm{m/s}$$ relative to the ground. The river flows at $$1.4\ \mathrm{m/s} in $$the $$-y$$ direction. What is the direction of the boat's velocity relative to the water?

Accepted Answer

$$16^\circ$$ above the $$+x$$ axis$

Question 11

Which of the following best describes the process for finding the velocity of one object relative to another using vectors?

Accepted Answer

Add the velocity of the first object relative to the second object to the velocity of the second object relative to the third object.

Question 12

A swimmer can swim at $$2.0\ \mathrm{m/s} in $$still water. If the river flows at $$1.0\ \mathrm{m/s}$$, what is the swimmer's speed relative to the ground if she swims directly upstream?

Accepted Answer

$$1.0\ \mathrm{m/s}$$

Question 13

A pilot wants to fly due north at $$200\ \mathrm{km/h}$$ relative to the ground. There is a wind blowing east at $$50\ \mathrm{km/h}. $$What must be the magnitude of the plane's velocity relative to the air?

Accepted Answer

$$\sqrt{200^2$ + 50^2$} = 206\ \mathrm{km/h}$$

Question 14

If $$\vec{v}_{PG} = \vec{v}_{PT} + \vec{v}_{TG}$$, and $$\vec{v}_{PT} = 0.24\ \mathrm{m/s}\ \hat{y}$$, $$\vec{v}_{TG} = 0.68\ \mathrm{m/s}\ \hat{x}$$, what is the angle $$	heta of$$ $$\vec{v}_{PG}$$ with respect to the $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{0.24}{0.68}ight) = 19^\circ$$

Question 15

A cyclist rides north at $$8\ \mathrm{m/s}$$ relative to the ground. A wind blows east at $$6\ \mathrm{m/s}$$ relative to the ground. What is the magnitude of the cyclist's velocity relative to the air?

Accepted Answer

$$10\ \mathrm{m/s}$$

Relative Motion and Vector Addition in Physics

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