Chapter 3: Kinematics in Two Dimensions

3.1: Displacement, Velocity, and Acceleration

Because displacement in two dimensions cannot be expressed solely in terms of x or y, displacement is often referred as r.
- r is the sum of a 2D displacement vector’s x and y components.
Following suit, average velocity becomes vₐᵥ = Δr / Δt while instantaneous velocity becomes
Values for acceleration remain the same.
- Average acceleration: aₐᵥ = Δv / Δt
- Instantaneous acceleration:

3.2: Equations of Kinematics in Two Dimensions

Recall how motion can be described in terms of displacement, acceleration, and initial/final velocity.
In 2D kinematics, all of these variables of motion must be expressed in terms of x and y separately.
Therefore, the equations remain the same, but include subscripts for x and y.
However, the time t remains the same for both directions.

Projectile motion: motion in a curved (parabolic) two-dimensional path
In projectile motion, the horizontal and vertical components are studied separately (like in the table)
Since there is no acceleration in the x direction, the horizontal velocity of a projectile will always remain constant.
On the other hand, the vertical acceleration will always equal the 9.80 m/s², the acceleration due to gravity
Projectiles also experience time and speed symmetry.

Consider the following scenario: A passenger is walking towards another passenger car at the back of a train moving across some tracks.
What is the velocity of the passenger?
To answer this question, you would have to consider the velocity of the passenger relative to the train and the velocity of the train relative to the ground.
- The total velocity of the passenger is the velocity of the passenger relative to the train plus the velocity of the train to the ground.
Generally, if one object is moving in the presence of another moving object, you have to add them together to get the total velocity.
Other examples:
- An airplane traveling within an air stream.
- A boat crossing a river against/with a current.

Because displacement in two dimensions cannot be expressed solely in terms of x or y, displacement is often referred as r.
- r is the sum of a 2D displacement vector’s x and y components.
Following suit, average velocity becomes vₐᵥ = Δr / Δt while instantaneous velocity becomes
Values for acceleration remain the same.
- Average acceleration: aₐᵥ = Δv / Δt
- Instantaneous acceleration:

Recall how motion can be described in terms of displacement, acceleration, and initial/final velocity.
In 2D kinematics, all of these variables of motion must be expressed in terms of x and y separately.
Therefore, the equations remain the same, but include subscripts for x and y.
However, the time t remains the same for both directions.

Projectile motion: motion in a curved (parabolic) two-dimensional path
In projectile motion, the horizontal and vertical components are studied separately (like in the table)
Since there is no acceleration in the x direction, the horizontal velocity of a projectile will always remain constant.
On the other hand, the vertical acceleration will always equal the 9.80 m/s², the acceleration due to gravity
Projectiles also experience time and speed symmetry.

Consider the following scenario: A passenger is walking towards another passenger car at the back of a train moving across some tracks.
What is the velocity of the passenger?
To answer this question, you would have to consider the velocity of the passenger relative to the train and the velocity of the train relative to the ground.
- The total velocity of the passenger is the velocity of the passenger relative to the train plus the velocity of the train to the ground.
Generally, if one object is moving in the presence of another moving object, you have to add them together to get the total velocity.
Other examples:
- An airplane traveling within an air stream.
- A boat crossing a river against/with a current.