Read the timing and height of a repeating motion

Set up the model

A useful answer starts with clear assumptions:

• The model repeats exactly for t ≥ 0.
• The sine argument is in radians.
• This is an ideal animation model, not measured motion data.

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.

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Read the timing and height of a repeating motion

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Question

Start with the question

A model marker has height h(t) = 3 + 2sin(πt/4) metres, where t is seconds. Find its height range, period, frequency, and first maximum after t = 0.

Before you calculate

Read what is known and what you need to find. Make a prediction before moving to the first calculation.

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1. 1

   Bound the heights

   −1 ≤ sin(πt/4) ≤ 1; 1 ≤ h(t) ≤ 5

   The midline is 3 m and the amplitude is 2 m. Moving two metres above or below the midline gives the extremes.

2. 2

   Find a complete cycle

   T = 2π/(π/4) = 8 s; f = 1/8 = 0.125 Hz

   The argument must increase by 2π to repeat. Angular frequency is π/4 rad/s, distinct from cycles per second.

3. 3

   Locate the first peak

   πt/4 = π/2; t = 2 s; h(2) = 5 m

   The motion starts on the midline and initially rises. A quarter of the eight-second period reaches its first maximum.

Common mistakes to catch

• The peak-to-peak height 4 m is twice the amplitude.
• Matching one height does not prove that a complete cycle has elapsed.

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.