Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Differentiate a polynomial position function.
- Solve for a zero velocity.
- Check the direction of motion before calculating distance.
Before you start
The power rule and solving a linear equation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A cart has position s(t) = 10t − t² metres during a modeled braking interval. When does its forward velocity reach zero, and how far has it travelled then?
Why this math matters
Position, velocity, and acceleration answer different questions: where, how fast in a signed direction, and how the velocity changes. Differentiation connects them. A formula can continue producing numbers after its intended physical interval ends, so a good mathematical answer also identifies where the model should stop being interpreted literally.
Set up the model
A useful answer starts with clear assumptions:
- The cart follows this exact classroom formula only until its first stop.
- t ≥ 0 is in seconds, and forward motion is the positive direction.
- No real braking or equipment design recommendation is implied.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
When does a model cart stop moving forward?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A cart has position s(t) = 10t − t² metres during a modeled braking interval. When does its forward velocity reach zero, and how far has it travelled then?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Differentiate position
v(t) = s′(t) = 10 − 2t m/s
The derivative of 10t is ten and the derivative of t² is 2t. At t = 3, the cart is still moving forward at four metres per second.
Find the first stop
10 − 2t = 0 ⇒ t = 5 s
Velocity is positive for 0 ≤ t < 5 and zero at five. This is the end of the braking interval specified by the model.
Evaluate travel and acceleration
s(5) − s(0) = 50 − 25 = 25 m; a(t) = v′(t) = −2 m/s²
Because motion stays forward before the stop, displacement equals distance travelled. Negative acceleration means the signed velocity decreases uniformly here.
The result
The cart first stops after 5 seconds, having travelled 25 metres. Its modeled acceleration is −2 m/s².
For t > 5, the same polynomial predicts backward motion. A real cart that remains stopped would need a different position rule after five seconds. Continuing the algebra is not the same as validating that physical prediction.
Common mistakes to catch
- Setting position equal to zero finds a location event, not a stopping time.
- Negative acceleration means slowing down only when velocity is positive in this coordinate system.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For s(t) = 12t − 2t², when is the first zero velocity?
Show a hint
Differentiate first, then solve 12 − 4t = 0.
Reveal answer and explanation
3 seconds
The velocity is 12 − 4t, which becomes zero at three seconds.
Practice 2
For the original cart, find velocity and acceleration at t = 4.
Show a hint
Use the first and second derivatives separately.
Reveal answer and explanation
2 m/s and −2 m/s²
v(4) = 10 − 8 = 2, while the constant second derivative remains −2.
Take the idea with you
Whenever you differentiate a physical model, write the new units and check the interval where its assumptions apply.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Why does a faster oscillation change the derivative?
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