Unit 4 · Lesson 3 of 5

Initial data selects an antiderivative.

Two carts can have the same velocity history while starting at different positions. The integration constant carries that missing information.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its position.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase initial position c? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Chosen positionReference C=0 · dashed

Initial position C2 mTime2 s

position

s(2)=2^2+2=6\,\mathrm m

All displayed position curves have the same velocity, yet different starting positions. Differentiation erases constants.

Assumptions, units, and limits

Integrating velocity leaves an arbitrary constant; an initial position fixes it.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

Indefinite integration gives a family of functions. An initial value fixes the constant. Verify the resulting function by differentiating and by checking the initial condition.

v(t)=2t,\quad s(t)=t^2+C

Keep the reasoning visible

\int2t\,dt=t^2+C

All displayed position curves have the same velocity, yet different starting positions. Differentiation erases constants.

What makes this model useful?

Integrating velocity leaves an arbitrary constant; an initial position fixes it.

The constant is not an optional decoration when solving an initial-value problem. It encodes the starting state.

A worked example

Calculate the starting scenario

Use initial position c = 2 m; time = 2 s.
$s(2)=2^2+2=6\,\mathrm m$
All displayed position curves have the same velocity, yet different starting positions. Differentiation erases constants. The checked quantity is position.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the position using initial position c 2 m, time 2 s. State the meaning of its sign and units.

Checked quantity: position · m. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Indefinite integration gives a family of functions. An initial value fixes the constant. Verify the resulting function by differentiating and by checking the initial condition.

Explore the visual solution

Chosen positionReference C=0 · dashed

Initial position C2 mTime2 s

position

s(2)=2^2+2=6\,\mathrm m

All displayed position curves have the same velocity, yet different starting positions. Differentiation erases constants.

Assumptions, units, and limits

Integrating velocity leaves an arbitrary constant; an initial position fixes it.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Indefinite integration gives a family of functions. An initial value fixes the constant. Verify the resulting function by differentiating and by checking the initial condition.

v(t)=2t,\quad s(t)=t^2+C

Calculate and explain

Apply the idea to a new scenario

For $C=-1$ m and $t=3$ s, find position under velocity $v=2$ t.

Checked quantity: position · m. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Indefinite integration gives a family of functions. An initial value fixes the constant. Verify the resulting function by differentiating and by checking the initial condition.

Explore the visual solution

Chosen positionReference C=0 · dashed

Initial position C-1 mTime3 s

position

s(3)=3^2+-1=8\,\mathrm m

All displayed position curves have the same velocity, yet different starting positions. Differentiation erases constants.

Assumptions, units, and limits

Integrating velocity leaves an arbitrary constant; an initial position fixes it.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Indefinite integration gives a family of functions. An initial value fixes the constant. Verify the resulting function by differentiating and by checking the initial condition.

v(t)=2t,\quad s(t)=t^2+C

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Chosen positionReference C=0 · dashed

Initial position C2 mTime2 s

position

s(2)=2^2+2=6\,\mathrm m

All displayed position curves have the same velocity, yet different starting positions. Differentiation erases constants.

Assumptions, units, and limits

Integrating velocity leaves an arbitrary constant; an initial position fixes it.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Indefinite integration gives a family of functions. An initial value fixes the constant. Verify the resulting function by differentiating and by checking the initial condition.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.