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 composition involving sine.
- Evaluate motion at a turning point.
- Use radian-based derivative formulas correctly.
Before you start
Sine and cosine at standard radian angles, plus polynomial derivatives.
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 model platform's vertical displacement is h(t) = 2 sin(3t) centimetres. What are its velocity and acceleration at t = π/6 seconds?
Why this math matters
An oscillating system combines an outer shape with an inner clock. The factor three changes how quickly the sine input turns, while the factor two changes displacement size. The chain rule keeps both effects. This is why a small, rapid vibration can have a substantial acceleration even when its displacement looks modest.
Set up the model
A useful answer starts with clear assumptions:
- The phase 3t is in radians, with angular frequency 3 radians per second.
- Displacement is measured from the equilibrium height.
- The ideal sinusoidal motion is a classroom model without damping.
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.
Why does a faster oscillation change the derivative?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A model platform's vertical displacement is h(t) = 2 sin(3t) centimetres. What are its velocity and acceleration at t = π/6 seconds?
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 the outer and inner functions
h′(t) = 2 cos(3t) × 3 = 6 cos(3t) cm/s
The derivative of sine is cosine when its angle is measured in radians. Multiplying by three accounts for the inner phase changing three times as fast as t.
Find acceleration
h″(t) = 6[−sin(3t)] × 3 = −18 sin(3t) cm/s²
Apply the chain rule again. Both the negative sign and the additional factor of three matter.
Evaluate at the top of the motion
3(π/6) = π/2; h = 2; h′ = 0; h″ = −18
At the highest displacement, the platform has zero instantaneous velocity but a downward acceleration. Zero velocity does not mean that motion will remain stopped.
The result
At π/6 seconds, velocity is 0 cm/s and acceleration is −18 cm/s².
The period is 2π/3 seconds because the phase must increase by 2π for a full cycle. Increasing angular frequency increases velocity amplitude linearly and acceleration amplitude quadratically.
Common mistakes to catch
- Forgetting the inner derivative gives a velocity amplitude of two instead of six.
- Using degree-based inputs in the standard sine derivative silently introduces the wrong scale.
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 h(t) = 5 sin(2t), find velocity at t = 0 using radians.
Show a hint
Differentiate and use cos 0 = 1.
Reveal answer and explanation
10 cm/s
h′(t) = 10 cos(2t), giving ten at zero.
Practice 2
For the original platform, what is its acceleration at t = 0?
Show a hint
Substitute into −18 sin(3t).
Reveal answer and explanation
0 cm/s²
sin 0 = 0, so acceleration is zero even though velocity is then 6 cm/s.
Take the idea with you
Look for an inner clock in any composed model, such as temperature depending on a changing position.
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: How does a changing flow rate fill a tank?
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.