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.
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!
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.
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.
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.
Unit 3 Assessment
- Receive a grade