### Unit 3: Kinematics in Two Dimensions

Most motion in nature follows curved paths rather than straight lines. Motion along a curved path on a flat surface or a plane is two-dimensional and thus described by two-dimensional kinematics. Two-dimensional kinematics is a simple extension of the one-dimensional kinematics covered in the previous unit. This simple extension will allow us to apply physics to many more situations and it will also yield unexpected insights into nature.

**Completing this unit should take you approximately 5 hours.**

Upon successful completion of this unit, you will be able to:

- add and subtract vectors;
- determine the components of a vector given its magnitude and direction and determine the magnitude and direction of a vector given its components; and
- separately analyze the horizontal and vertical motions in projectile problems.

### 3.1: Introduction to Kinematics in Two Dimensions using Vectors

Two-dimensional kinematics is surprisingly easy. They are similar to one-dimensional problems, due to our coordinate system. Notice that coordinate systems have perpendicular axes, and motion along the two axes is independent from each other. So, the physics or math that helps us solve for an object's motion in the x−direction does not influence its motion in the y−direction. Solving for two-dimensional motion is like solving for one-dimensional motion twice!

Read this text, which provides a general overview of the concept of kinematics in two dimensions.

To solve two-dimensional kinematic problems, we first need to understand how two-dimensional motion is represented and how to break it up into two one-dimensional components. We also need to understand vectors. A vector is a quantity that has both a magnitude (amount) and direction. Often in texts, vectors are denoted by being bolded or having a small arrow written above the vector name.

For example, a vector called A can be written as A or as . The magnitude, or amount, of the vector A equals the value of A. And, the direction of A is given by some other notation usually accompanying the value. We can think of vectors as arrows, with the length being the magnitude of the vector and the arrow pointing in the direction of the vector.

Vectors are often notated like this: . The denotes that the magnitude is the part of the vector that protrudes the x-axis. Similarly, the denotes that the magnitude is the part of the vector that protrudes the y-axis.

So, for example, the vector extends down the x-axis three units while extending up the y-axis five units. This notation is called "component form" and is a preferred way of representing vectors.

Another way of representing vectors is by denoting their magnitude and direction. For example, we can denote the vector A, shown in the previous paragraph, also as 5.83 units 59 degrees from the x-axis. Notice that we need to specify that the direction has an angle with respect to the x-axis. Not only do we need an angle, but we also need a reference point from which the angle spawns. Generally, the x-axis is a convenient choice. We call this notation the magnitude-direction form.

This video discusses vector notation. Note that they use engineering notation, which replaces x-hat with i-hat and y-hat with j-hat. The meanings are the same despite these changes.

### 3.2: Adding and Subtracting Vectors

When adding or subtracting vectors, we can follow many of the rules we learned in math class about non-vector numbers. Vector addition follows the commutative property, which means the order of addition does not matter. Vector addition also follows the associative property, which means it does not matter which vector is first when vectors are being added.

One way to add or subtract vectors is to do so graphically. The graphical method for adding and subtracting vectors is called the head-to-tail method. When adding vectors using this method, use these steps:

1. Draw the first vector starting from the tail, or starting point of the vector, to its head, or ending point (arrow) of the vector.

2. Begin the second vector by putting its tail at the head of the first vector.

3. Finally, draw a line from the tail of the first vector to the head of the second vector.

The vector that results is the resultant vector, or the solution to the vector addition problem. To determine the magnitude of the resultant vector, measure it with a ruler. To determine the direction of the resultant vector, use a protractor to determine the angle from one of the axes. When subtracting vectors graphically, consider the vector that is being subtracted as negative. That means the direction of the vector being subtracted is flipped so it points in the opposite direction. The head-to-tail process is the same as it is for addition.As you read, pay attention to the worked examples: using the head-to-tail method to add multiple vectors in Example 3.1 and using the head-to-tail method to subtract vectors in Example 3.2.

This lecture accompanies what you just read.

### 3.3. Adding Vectors Analytically: Determining the Components, Magnitude, and Direction of a Vector

We can also use analytical methods to add and subtract vectors. Analytical methods use trigonometry to solve vector addition and subtraction. While we still use arrows to represent vectors, analytical methods reduce the measurement errors that can occur with graphical (head-to-tail) methods.

Read this text, which explains how we need to resolve vectors into their component vectors in the x-y coordinate system when using analytical methods to solve vector problems. See Figure 3.26 for an example of a vector that has been resolved into its x and y components. Here, the vector A has a magnitude A and an angle 𝛳. We can break the vector down into two components: Ax and Ay. We know that . However, we must use trigonometry to determine how the scalar or magnitude part of each vector relates to one another. You do not need to know the inner workings of trigonometry to deal with vectors analytically, but you need to understand their basic functions and know how to input a sine and cosine function into a calculator. The magnitudes of the component vectors relate to the resultant vector this way:

See Figure 3.27 for an example of a vector that has been resolved into its component vectors and shows the magnitudes of the component vectors. Note that these equations work if you want to find the angle at the bottom-left of the right triangle in Figure 3.27. If you want to find the angle at the top of the triangle, you would use the sine function for and the cosine function for .

The general rule is: use the sine function for components of the vector that are opposite of the triangle as the angle, and use the cosine function for components of the vector that are adjacent on the triangle to the angle. For example, in Figure 3.27, you can see that the y-component of the vector is on the opposite side of the triangle as the angle so the sine function is used to find the y-component, as in the previous equation.

Sometimes you are given the component vectors and need to determine the magnitude and angle of the resultant vector. To do this, we again use the trigonometry of right triangles:

The angle obtained by using the tangent equation is such that the opposite component of the vector is the y-component, and the adjacent component is the x-component. Also, pay attention to the example of a resultant vector calculated from its component vectors in Figure 3.29.

This lecture accompanies what you just read.

This video discusses how vectors are represented as components in the x- and y-axes.

This video gives an example of how we use components representing vectors in the x- and y-axes in two-dimensional kinematic problems.

This video describes how to convert vectors from magnitude-direction form into component form, and vice versa.

This video demonstrates how to add vectors using the graphical and analytical methods.

### 3.4: Projectile Motion and Trajectory

Often, physics problems occur on the surface of the Earth, such as footballs being kicked, rockets being fired, and daredevils riding their motorcycles off cliffs. This means that the y-component of these two-dimensional motions involve acceleration pointing downward while the x-component does not have any acceleration. We call these types of motion projectile motion.

We define projectile motion as the motion of a thrown object that only feels the acceleration of gravity. The projectile is the object being thrown; the trajectory is the path the object takes when it is thrown.

We need to use the kinematic equations we learned in Unit 2 of this course to calculate projectile motion, for each of the two-dimensions separately. Note that we assume there is no air resistance when we perform projectile motion calculations – so gravity is the only force acting on the projectile.This reading discusses how vectors are represented as components in the x- and y-axes.

This lecture accompanies what you just read.

Watch this video to see another way to solve for the time of a projectile in air. Since time transcends the x- and y-components of a trajectory, it is important to know how to calculate time so you can use it to connect motion in the two dimensions.

Watch this video to learn how to solve for a horizontally-launched projectile, with no initial y-component to velocity.

Watch this video on how to solve kinematic equations for the x and y directions of motion, using the same procedures as for a horizontally-launched projectile.

Watch this video for another example of projectile motion and how to solve for quantities using kinematic equations for both dimensions.

Watch these two videos on how to solve for the total final velocity of the projectile at the end of its path. Note that the presenter makes a small mistake which they correct in the second video.

Watch this video on how to solve projectile motion problems using kinematic equations for trajectories that start at an incline.

### Unit 3 Assessment

- Receive a grade
Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment