Motion in a Plane: Principles & Calculations

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

Motion in a plane includes linear motion, rotational motion, and projectile motion. This lesson will focus on two-dimensional, linear motion of a single object, and linear motion of two objects moving relative to each other. In both scenarios unit-vectors will be utilized to organize the displacements, velocities, and accelerations of the object(s); and techniques will taught that help to solve these types of kinematics problems.

Planar motion

Motion can occur in one-dimension, in two-dimensions or in three-dimensions. Planar motion is when an object moves in two directions at the same time. For example, walking North and East simultaneously (Northeast) at a rapid rate of speed to get away from someone begging you to do their physics homework for them is planar motion. The use of a coordinate system is critical when analyzing motion in a plane. We use unit-vectors to designate direction along the axes. All unit-vectors have a magnitude of one and point in the positive direction of their axis. The symbol for a unit vector is a letter with a caret symbol over it. The x-direction unit vector is î, also known as i-hat. We use j-hat for the y-direction and k-hat for the positive z-direction.


3-D coordinate system showing unit-vectors.
ijkvectors

Linear Motion of a Single Object in Two-dimensions

Let's say we have a mass that started at the origin and three seconds into its motion reached point Q located at 4 meters along the negative x-axis and 3 meters up the positive z-axis. We would represent this mass's position at t = 3 s in three-dimensions (x,y,z) as (-4,0,3). Figure 1 shows point Q with its displacement vector (red arrow) from the origin to point Q in a three-dimensional grid. A three-dimensional grid is used to illustrate the two-dimensional plane that the mass is in. Our mass is in the x-z plane.


Figure 1. The displacement vector is written in unit vector notation.
2dposition

The mass at point Q was initially traveling at velocity


velocity_vector

and suddenly at position coordinates (-4 m, 0 m, 3 m) it experienced a force giving it an x-direction acceleration.


acceleration_vector

We could only measure the x-direction acceleration, and the mass eventually ended up back at the origin. What was the mass's acceleration in the z-direction?

Since we are dealing with motion in a plane we must treat the motion in each direction separately. It is best to start with a kinematic equation containing the variables that you have been given and the variable you are asked to determine.


equationset1

Notice that there are two variables in the final equation in z-direction; t and az. We cannot solve for the acceleration in the z-direction (what we are asked to solve) without knowing the time. Time is the only variable in x-direction so if we solve for time in the x-direction we can plug it into the z-direction equation to determine the z-acceleration. Time is the link between both directions because it is scalar. Let's break out the quadratic equation to solve for time.


This is the only real root of this quadratic because time can not be negative.
quadratic

Plugging this time into the z-direction equation gives us the z-acceleration.


z_axis_acceleration

Relative Motion

One-dimensional Linear Motion Problem

In the previous scenario we dealt with a single object traveling in the x-z plane. What happens if we have two objects traveling in the same plane relative to each other? How does each object see the other object's motion? To get a grasp on what relative means let's start with a relative motion scenario in one-dimension.

A car is traveling due East on a straight road at


velocity_car_1d

and a motorcycle is ahead of the car on the same road traveling at


velocity_motorcycle_1d

What is the velocity of the motorcycle relative to the car? In other words, to a passenger in the car, how fast is the motorcycle traveling? To solve this we will have to pick a reference point that is common to both objects; in this case it will be the ground. We will make our velocities more specific by adding another letter in the subscripts representing the ground.


vel_c_g

These subscripts are read the velocity of the car relative to the ground and the velocity of the motorcycle relative to the ground. We are going to add these velocities using a special organization of the subscripts.


vec_setup

The reference point is always sandwiched between the first and second letters of what is being asked about. We are asked for the velocity of the motorcycle relative to the car (vmc). After the equals sign we write a velocity whereby we take the first letter m and put the reference point g after it (vmg). This is added to a velocity where the g is placed in front of second letter c (vgc). Notice, though, that we don't have (vgc), we have (vcg). Since these letters are reversed in the subscript we can reverse our vector by making it negative.


v_mc_solution


motorcycle_car_sketch

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