4 Chapter 4: Two Dimensional Motion
Section 4.1: Motion in Two and Three Dimensions
We began in the last chapter with the most simple kind of motion – motion in a straight line. Now we’re going to generalize this motion from a single dimension (up/down, left/right, or forward/back) to combinations of all of these dimensions. We’ll use the same three kinematics equations to describe this motion, but now instead of restricting ourselves to one direction at a time, we’ll use a combination of straight-line motion in two or three dimensions at once.
The important thing to realize here is that it’s the same motion we saw in the last chapter – we’re still moving at a constant acceleration, but now we might have different motion in the direction than we did in the direction. It’s vital to remember that changing your velocity, position, or acceleration does not change anything about your position, velocity, or acceleration in the and directions. The three directions are orthogonal to each other – changing one does not change the others.
Because we will now be talking about motion in the , , and sometimes direction, we’re going to introduce a new way of writing position that doesn’t limit us to constantly referring to the direction. The new position vector is written as:
The three equations above are all equivalent ways of writing the position vector .
As a result, the displacement vector, the straight-line displacement between any two locations in three dimensions, is given by:
Now that we have a position vector, we can re-define our velocity vector in terms of the position vector.
The three equations above are equivalent statements of the velocity vector. Finally, we can do the same with the acceleration vector .
Textbook 4.1: Displacement and Velocity Vectors
Textbook 4.2: Acceleration Vector
Section 4.2: Projectile Motion
An object subject to a constant acceleration at an angle relative to its motion will move in a two-dimensional path known as its trajectory. Most of the objects we encounter will be subject to the constant acceleration due to the gravitational force. Any object subject to gravity is known as a projectile.
Two-dimensional kinematics works the same way as one-dimensional kinematics, but now acting in two orthogonal (perpendicular) directions. The important points to remember here are (1) that the and directions are orthogonal, which means they are perpendicular and that one does not rely on the other (a change in does not mean you have aprojectile change in , and (2) that we will have a constant acceleration acting on our object, in either the direction, the direction, or in some combination of the and directions. Our derivation of the kinematics equations depended on a constant, non-changing acceleration.
We again use the one-dimensional kinematics equations, but now we have one set in the direction and one set in the direction. The only variable they have in common is the time measured in seconds.
Section 4.3: Kinematics Problem-Solvi