HomeConcept libraryMechanicsMotion and Circular Motion

MechanicsIntroStarter track

Concept module

Vectors and Components

Rotate and scale a live vector, decompose it into horizontal and vertical parts, and watch those components drive the same straight-line motion and geometry.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 4.00 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s4.00 s

Vectors and components

Drag the 1 s vector handle to set magnitude and angle.

Graphs

Switch graph views without breaking the live stage and time link.

Position path

Shows the straight-line path traced by the moving point in the plane.

horizontal displacement (m): -48 to 48vertical displacement (m): -48 to 48

Path

Hover or scrub to link the graph back to the stage.horizontal displacement (m) / vertical displacement (m)

Controls

Adjust the physical parameters and watch the motion respond.

Magnitude

v

8 m/s

Sets the total vector size and the one-second step length.

Angle

θ

35°

Positive angles point above the x-axis and negative angles point below it.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Graph readingPrompt 1 of 2

Notice that the path graph and the stage are the same straight-line motion shown in two representations.

Try this

Hover or scrub the path graph and watch the moving point jump to the matching place in the stage.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

v

Vector magnitude

8 m/s

Sets the total size of the vector, so both components and the travelled distance scale with it when the angle stays fixed.

Graph: Position pathGraph: Position componentsGraph: Vector componentsOverlay: Component guidesOverlay: 1 s reference step

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompt as a compact guide while you rotate or scale the vector. The best prompt should point you at a pattern the current stage or graph actually shows.

Graph readingPrompt 1 of 2

Graph: Position path

Notice that the path graph and the stage are the same straight-line motion shown in two representations.

Try this

Hover or scrub the path graph and watch the moving point jump to the matching place in the stage.

Why it matters

It keeps the geometry and the graph linked instead of treating the plot as a separate diagram.

Graph: Position pathOverlay: Component guides

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Component guides

Projects the current point onto the horizontal and vertical axes.

What to notice

The component legs are the exact horizontal and vertical distances at the current time.

Why it matters

It turns the diagonal motion into the right triangle that the equations describe.

Control: MagnitudeControl: AngleGraph: Position pathGraph: Position componentsEquation

v

Equation

θ

Challenge mode

Use the vector controls and inspect-time rail as a compact component-matching tool instead of a passive diagram.

0/2 solved

MatchCore

1 of 3 checks

Equal components

Build a vector whose horizontal and vertical components are nearly the same size. Keep the component graph open so the match is visible in the real readout.

Graph-linkedGuided start2 hints

Suggested start

Use the component graph and guides together while you adjust the angle.

Pending

Open the Vector components graph.

Position path

Matched

Keep the Component guides visible.

Pending

Make the component mismatch smaller than

0.2 m/s

1.96 m/s

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the point is at x = 0 m and y = 0 m. The vector magnitude is 8 m/s at 35°, so v_x = 6.55 m/s points to the right and v_y = 4.59 m/s points upward.

Equation detailsDeeper interpretation, notes, and worked variable context.

Horizontal component

Projects the vector onto the horizontal axis.

v_{x} = v cos (θ)

v

Vector magnitude 8 m/s

θ

Angle 35°

Vertical component

Projects the vector onto the vertical axis.

v_{y} = v sin (θ)

v

Vector magnitude 8 m/s

θ

Angle 35°

Horizontal position

Tracks how far the point moves horizontally when the horizontal component stays constant.

x (t) = v_{x} t = v cos (θ) t

v

Vector magnitude 8 m/s

θ

Angle 35°

Vertical position

Tracks how far the point moves vertically when the vertical component stays constant.

y (t) = v_{y} t = v sin (θ) t

v

Vector magnitude 8 m/s

θ

Angle 35°

Resultant magnitude

Reconstructs the original vector from its perpendicular components.

v = v_{x}^{2} + v_{y}^{2}

This relation works because the component legs are perpendicular.

The same geometry later reappears in velocity, displacement, and force problems.

v

Vector magnitude 8 m/s

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 1 of 30 / 3 complete

Motion and Circular Motion

Next after this: Projectile Motion.

1. Vectors and Components2. Projectile Motion3. Uniform Circular Motion

This concept is the track start.

See next concept

Short explanation

What the system is doing

A two-dimensional vector does not need a special diagonal rule. You can project it onto the horizontal and vertical axes, then track those perpendicular parts with ordinary algebra and geometry.

This module uses a constant velocity vector so the component idea stays visible. The same magnitude and angle determine the one-second reference vector, the moving point, the component graphs, and the straight-line path, which is why the decomposition carries directly into later mechanics.

Key ideas

01The horizontal and vertical components are projections of the same vector, not separate pushes you invent afterward.

02At fixed magnitude, changing angle redistributes the vector between horizontal and vertical parts.

03If the components stay constant, the resulting motion stays on one straight path while the x(t) and y(t) graphs remain linear.

Live component checks

Solve the exact state on screen.

Solve the current vector directly from the live controls. The same magnitude and angle drive the stage, the graphs, and these substitutions, and the time-based example follows the current inspected time unless you freeze it.

Live valuesFollowing current parameters

For the current vector, what horizontal and vertical components does $v$ have?

v

Vector magnitude

8 m/s

θ

Angle

35 °

1. Identify the component relations

Use

v_{x} = v cos (θ)

and

v_{y} = v sin (θ)

2. Substitute the live magnitude and angle

v_{x} = 8 cos (3 5^{\circ})

and

v_{y} = 8 sin (3 5^{\circ})

3. Compute each component

That gives

v_{x} = 6.55 m/s

and

v_{y} = 4.59 m/s

Current components

v_{x} = 6.55 m/s, v_{y} = 4.59 m/s

Both components are positive, so the vector points up and to the right in the first quadrant.

Decompose it

Keep the vector magnitude fixed and rotate the direction closer to the vertical axis. What must happen to the horizontal and vertical components, and how does the straight-line path change?

Prediction prompt

Predict which component grows, which shrinks, and whether the path becomes flatter or steeper before you test it.

Check your reasoning

The vertical component grows in magnitude while the horizontal component shrinks, so the path tilts closer to vertical.

At fixed magnitude, rotating the vector redistributes the same total length between the component legs. The resultant stays the same size, but the right-triangle geometry changes.

Common misconception

A diagonal vector needs its own separate rule, so the components are only rough approximations.

The components are exact perpendicular projections of the same vector, which is why the Pythagorean and trigonometric relations recover the original magnitude and angle.

A negative component does not mean the vector got smaller. It only means that piece points leftward or downward relative to the chosen axis.

Quick test

Compare cases

Question 1 of 4

Use the stage, component graph, and position graph together. These checks are about explaining what the vector does, not just naming formulas.

Two vectors have the same magnitude. One points at $3 0^{\circ}$ and the other at $6 0^{\circ}$ . Which comparison is correct?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a coordinate plane with a draggable vector anchored at the origin and a point moving in the vector direction over time. Optional overlays show the angle marker, a one-second reference step, and the horizontal and vertical component guides.

Changing the magnitude or angle immediately updates the path, the position graphs, and the constant component graph so the same vector decomposition stays synchronized across every representation.

Graph summary

The path graph is a straight line through the plane because the components stay constant. Hovering or scrubbing the graph moves the stage point to the same place on that line.

The position graph shows linear x(t) and y(t) trends, while the component graph shows flat vx and vy lines because the vector components do not change with time in this model.

Build toward mechanics

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Turning-effect bridgeMechanics

Vectors and Components

See each variable before you move it.

Component guides

Equal components

Motion and Circular Motion

What the system is doing

Solve the exact state on screen.

For the current vector, what horizontal and vertical components does $v$ have?

Two vectors have the same magnitude. One points at $3 0^{\circ}$ and the other at $6 0^{\circ}$ . Which comparison is correct?

Keep moving through the concept map.

Torque

Momentum and Impulse

Projectile Motion

Vectors and Components

See each variable before you move it.

Component guides

Equal components

Motion and Circular Motion

What the system is doing

Solve the exact state on screen.

For the current vector, what horizontal and vertical components does v have?

Two vectors have the same magnitude. One points at 30∘ and the other at 60∘. Which comparison is correct?

Keep moving through the concept map.

Torque

Momentum and Impulse

Projectile Motion

For the current vector, what horizontal and vertical components does $v$ have?

Two vectors have the same magnitude. One points at $3 0^{\circ}$ and the other at $6 0^{\circ}$ . Which comparison is correct?