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What we found:

A rock is kicked horizontally at \( 15 \) \( \text{m/s} \) from a hill with a \( 45^\circ \) slope. How long does it take for the rock to hit the ground?

Tarzan is above the floor of the jungle on a limb as shown in the picture below. He swings out on a vine of length \( 15 \) \( \text{m} \) that initially makes an angle of \( 32^{\circ} \) with the vertical. He releases his grip on the vine when he is at the lowest point of the swing. At this lowest point, he is \( 9.0 \) \( \text{m} \) above the ground. How far horizontally did he travel from when he first started his swing (distance \( d \) in the picture)?

A pitcher can throw a ball at a velocity of \( 125 \) \( \text{km/h} \) straight ahead. If he throws the ball straight when a cross-wind is blowing at \( 28 \) \( \text{km/h} \) to the left, what will the magnitude of the ball’s resultant velocity be? By what angle will the direction of the ball be off?

The components of vector \( \vec{E} \) are: \( E_x = -34.8 \) and \( E_y = -23.6 \). What is the measure of the angle, in degrees, formed by vector \( \vec{E} \) and the \( +x \)-axis?

The magnitude of vector \( \vec{A} \) is \( 8.6 \). The vector lies in the fourth quadrant and forms an angle of \( 37^{\circ} \) with the \( x \)-axis. What are the components of vector \( \vec{A} \)?

Find the \( X \) and \( Y \) components of the following vectors.

Vectors \( \vec{A} \) and \( \vec{B} \) originate from the same point on a set of perpendicular \( x \)– and \( y \)-axes as shown in the accompanying figure. Vector \( \vec{A} \) has magnitude \( 6.3 \) and is directed \( 23^{\circ} \) to the left of the positive \( y \)-axis. Vector \( \vec{B} \) has magnitude \( 5.7 \) and is directed \( 34^{\circ} \) above the positive \( x \)-axis. A third vector is defined by \( \vec{C} = \vec{B} – \vec{A} \). What is the magnitude of \( \vec{C} \)?

Challenge: Listed below are \( 4 \) distance vectors.

A: \( 15 \) \( \text{m} \) at \( 60^{\circ} \) above the \( +x \)-axis
B: \( 10 \) \( \text{m} \) east
C: \( 25 \) \( \text{m} \) south
D: \( 50 \) \( \text{m} \) at \( 28^{\circ} \) South of West

Using the vectors above, perform the following operations:

The river bank is \( 30 \) \( \text{km} \) South and \( 9 \) \( \text{km} \) West of the school. If a bird flies directly from the school to the bank, what is the magnitude and direction of its displacement vector?

Find the resultant vector (magnitude AND direction) given the following components:

How fast is this plane traveling? (What is the magnitude of the plane’s velocity?) What is the angle \( \Theta \)? The plane’s velocity components are \( 120 \) \( \text{m/s} \) in the positive \( x \)-direction and \( 80 \) \( \text{m/s} \) in the positive \( y \)-direction, as shown in the diagram.

A sailor heads out on a lake at \( 2.5 \) \( \text{m/s} \) with a bearing of \( 30^{\circ} \) north of east. A wind headed directly south-west is blowing at \( 1.0 \) \( \text{m/s} \). What is the resultant velocity of the sailor? (magnitude and direction)

The components of vectors \( \vec{A} \) and \( \vec{B} \) are given as follows:
\( A_x = 5.1 \) \( B_x = -2.6 \)
\( A_y = -5 \) \( B_y = -4.3 \)

What is the magnitude of vector sum \( \vec{A} + \vec{B} \)?

Three students are pulling on a bag of skittles. Each applies a horizontal force. Student \( 1 \) pulls eastward with a force of \( 170 \) \( \text{N} \). Student \( 2 \) pulls southward with a force of \( 100 \) \( \text{N} \). Student \( 3 \) pulls with a force of \( 200 \) \( \text{N} \) at an angle of \( 20^{\circ} \) west of north. What is the net force exerted on the bag of skittles by the three students?

A \( 100 \) \( \text{N} \) lawnmower is being pushed with a handle that makes a \( 37^\circ \) angle to the ground. There is \( 10 \) \( \text{N} \) of friction between the mower and the ground. What force must be applied to the handle of the mower to get it to accelerate at \( 3 \) \( \text{m/s}^2 \)?

A \( 20 \) \( \text{N} \) force due north and a \( 20 \) \( \text{N} \) force due east act concurrently on an object, as shown in the diagram to the right. What additional force is needed to bring the object into a state of equilibrium?

An object is moving to the west at a constant speed. Three forces are exerted on the object. One force is \( 10 \) \( \text{N} \) directed due north, and another is \( 10 \) \( \text{N} \) directed due west. What is the magnitude and direction of the third force if the object is to continue moving to the west at a constant speed?

An object travels \( 30 \) \( \text{m} \) to the right in \( 5.0 \) \( \text{s} \), then \( 10 \) \( \text{m} \) to the left in \( 2.0 \) \( \text{s} \). It remains at rest for another \( 3.0 \) \( \text{s} \). What are its average speed and average velocity during the full \( 10 \) \( \text{s} \)?

A student walks \( 3 \) \( \text{m} \) east, then \( 4 \) \( \text{m} \) west in \( 7 \) \( \text{s} \). What is their displacement and average velocity?

When we refer to an object’s speed, we’re talking about:

What does displacement mean in the context of motion?

What does an object’s velocity describe?

An airplane is traveling \( 900. \) \( \text{km/h} \) in a direction \( 38.5^\circ \) west of north.

Vector \(\vec{A}\) is \(44.0\) units at \(28.0^{\circ\}) above the \(+x\)-axis, vector \(\vec{B}\) is \(26.5\) units at \(56.0^\circ\) above the \(-x\)-axis, and vector \(\vec{C}\) is \(31.0\) units along the \(-y\)-axis. Determine the resultant (sum) of the three vectors.

A skier is accelerating down a \( 30.0^{\circ} \) hill at \( 3.80 \) \( \text{m/s}^2 \).

Vector \( V_1 \) is \( 6.0 \) units long and points along the negative \( y \) axis. Vector \( V_2 \) is \( 4.5 \) units long and points at \( +45^{\circ} \) to the positive \( x \) axis.

An object is moving to the west at a constant speed. Three forces are exerted on the object. One force is \( 10 \) \( \text{N} \) directed due north, and another is \( 10 \) \( \text{N} \) directed due west. What is the magnitude and direction of the third force if the object is to continue moving to the west at a constant speed?

The magnitude of a component of a vector must be

You are adding vectors of length \( 20 \) and \( 40 \) units. Which of the following choices is a possible resultant magnitude?

A boat can row across a still \( 1 \, \text{km} \) wide river at a maximum speed of \( 5 \, \text{km/hr} \). If a current of \( 4 \, \text{km/hr} \) flows east as you try to directly cross the river, how long would it take?

You are piloting a small plane, and you want to reach an airport \(450 \, \text{km}\) due south in \(3.0 \,\text{hours}\). A wind is blowing from the west at \(50.0 \,\text{km/h}\). What heading and airspeed should you choose to reach your destination in time?

While Santa was delivering presents to the children of Nashville, Tennessee he experienced a strong wind perpendicular to his motion.

Gregory was walking through the halls of the school when he realized that he was walking in perpendicular directions and he could easily calculate his displacement using the incredibly useful techniques he learned in physics. He recognized that he walked \(12.5\ \text{m}\) left and then \(18.9\ \text{m}\) down. How far must he walk to the right so that his resultant displacement is \(20.1\ \text{m}\)?

An object travels along a path shown above, with changing velocity as indicated by vectors A and B. Which vector best represents the net acceleration of the object from time [katex] t_A [/katex] to [katex] t_B[/katex]?

A person is standing at the edge of the water and looking out at the ocean. The height of the person’s eyes above the water is \( h = 1.8 \, \text{m} \), and the radius of the Earth is \( R = 6.38 \times 10^6 \, \text{m} \). How far is it to the horizon (in meters)? In other words, find the distance \( d \) from the person’s eyes to the horizon. (Note: At the horizon, the angle between the line of sight and the radius of the Earth is \( 90^\circ \).)

Two racing boats set out from the same dock and speed away at the same constant speed of \(101 \, \text{km/h}\) for half an hour (\(0.5 \, \text{hr}\)). Boat 1 is headed \(27.6^\circ\) south of west, and Boat 2 is headed \(35.3^\circ\) south of west, as shown in the graph above. During this half-hour calculate:

A seagull first flies \( 160 \, \text{m} \) North, then heads \( 120.65 \, \text{m} \) at \( 18.43^\circ \) North of West. After you land:

Determine the sum of the three vectors given below. Give the resultant \( \vec{R} \) in terms of: (a) Vector components (b) Resultant vector Vectors: \( \vec{A} = 26.5 \, \text{m} \) @ \( 56^\circ \) NW \( \vec{B} = 44 \, \text{m} \) @ \( 28^\circ \) NE \( \vec{C} = 31 \, \text{m} \) South