Describing Motion in Two Dimesions

Motion in Two Dimensions (Notes)

What is two-dimensional motion?

• Two-dimensional motion means moving in two directions at the same time.

• Usually involves:

  • Horizontal motion (forward/backward)

  • Vertical motion (up/down)

Examples

• Snowboarder going downhill:

  • Moves forward and downward

• Person walking up a ramp:

  • Moves forward and upward

• Ball rolling down an incline:

  • Moves forward and down

All of these are examples of motion in two dimensions.

Breaking motion into components

• Any straight-line motion can be split into two components:

  • Horizontal component

  • Vertical component

• These components are treated independently.

Use of vectors

• Vectors are used to represent:

  • Position

  • Velocity

  • Acceleration

• Vectors show both:

  • Magnitude (how much)

  • Direction (which way)

Motion of an Object in Two Dimensions (Notes)

Coordinate system

• Motion is described using a Cartesian coordinate system:

  • x-axis → horizontal direction

  • y-axis → vertical direction

One-dimensional motion

• Motion along only one axis:

  • Horizontal motion (x-axis) or

  • Vertical motion (y-axis)

• Example:

  • A car moving straight forward

  • An object falling straight down

Two-dimensional motion

• Motion that occurs at an angle to the x-axis and y-axis.

• The object moves in both directions at the same time:

  • Horizontal (x)

  • Vertical (y)

Components of motion

• An angled motion can be split into two components:

  • x-component (horizontal)

  • y-component (vertical)

• The angle of motion determines how much of the motion is in each direction

Components of Two-Dimensional Motion (Notes)

x- and y-components

• Motion can be broken into:

  • x-component → horizontal motion

  • y-component → vertical motion

• Example (Eric downhill):

  • x-component = forward motion

  • y-component = downward motion

Sign conventions (very important)

• x-axis:

  • Right of the origin → positive

  • Left of the origin → negative

• y-axis:

  • Above the origin → positive

  • Below the origin → negative

Use of vectors

• Vectors describe:

  • Position

  • Velocity

  • Acceleration

• Vectors are useful because they:

  • Show magnitude and direction

  • Separate motion into x and y components

Independence of motion

• The x and y components are independent.

• A change in motion along one axis:

  • Does not affect motion along the other axis

• Example:

  • Horizontal speed stays constant even if vertical motion changes due to gravity.

The MASTER MENTAL MAP (use this every time)

From x–y graph:

• Straight line → constant direction

• Curved line → changing direction

• Longer stretch → larger displacement component

From x–t or y–t graph:

• Straight line → constant velocity

• Curved line → acceleration present

From velocity:

• Constant → acceleration ≈ 0

• Changing → acceleration ≠ 0

1⃣ Displacement as a vector (what x, y, and r mean)

• Eric moves:

  • x forward (horizontal)

  • y downward (vertical)

• These are the components of his displacement.

Together, they form a right triangle:

• One leg = xxx

• Other leg = yyy

• Hypotenuse = rrr

👉 r is the total displacement, meaning:

That’s why:

4⃣ What the x–t and y–t graphs are telling you

Linear x vs t graph (negative slope)

• x changes by the same amount each second

• Motion in x is constant velocity

• Negative slope → moving left (or backward relative to origin)

Linear y vs t graph (negative slope)

• y changes by the same amount each second

• Motion in y is also constant velocity

• Negative slope → moving downward

👉 Together this means:

• Eric’s velocity is constant in both directions

• He is sliding smoothly down a straight hill

• Acceleration is approximately zero

5⃣ Why the slopes matter

The slopes of the x–t and y–t graphs determine:

• How much of the motion is horizontal vs vertical

If:

• ∣slopex∣>∣slopey∣|\text{slope}_x| > |\text{slope}_y|∣slopex​∣>∣slopey​∣

Then:

• More motion is along x than y

• The hill is shallow, not steep

That’s why:

6⃣ The big picture (tie it all together)

Here’s the full logic chain:

1. Motion happens in x and y simultaneously

2. x and y form a right triangle

3. Pythagorean theorem gives total displacement rrr

4. Tangent gives the incline angle θ\thetaθ

5. Linear x–t and y–t graphs → constant velocity

6. Constant velocity → acceleration ≈ 0

7. The hill’s slope controls how much motion is horizontal vs vertical

cosine hugs the angle and sine stands across it.