LEARN · EXPLAIN · REVISE

A current moves east at 3 m/s. A boat moves at 5 m/s relative to water. Choose its velocity to travel due north across a 20 m river, and find the time.

Read the idea, work independently, then explain what changed.

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高一必修第二册(A版).pdf · 6.1 · PDF 9 / printed page 2

TOPIC 01

Vectors as displacements

Translate, add and scale arrows; distinguish a journey from its net displacement.

What you will be able to explain

Represent a displacement independently of its starting point.
Use head-to-tail addition and absolute scaling of lengths.
Choose a velocity that meets a directional constraint.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Your turn

A current moves east at 3 m/s. A boat moves at 5 m/s relative to water. Choose its velocity to travel due north across a 20 m river, and find the time.

\text{east}=x,\ \text{north}=y

Skills and prerequisite lessons

Scope: JM02

Working and explanation

BUILD THE REASONING

Hint 1

Cancel the current horizontally.

Hint 2

Use the fixed speed to obtain the north component.

Worked solution

Due north requires the boat to supply −3 horizontally.
$v_x=-3$
Choose the positive vertical root.
$v_y=\sqrt{25-9}=4$
Ground velocity is (0,4), so time is distance over speed.
$t=20/4=5\text{ s}$

Boat velocity (−3,4) m/s; crossing time 5 s.

Checks and common pitfalls: Pointing the boat due north does not produce a due-north ground path.

Reasoning checklist · self / teacher assessment

Boat velocity (−3,4) m/s; crossing time 5 s.
Pointing the boat due north does not produce a due-north ground path.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Prediction and diagnosis	Compare two translated arrows and vote on equality.
8	Concept derivation	Derive head-to-tail addition and negative scaling using a closed route.
10	Guided application	Work the translated-arrow example; pairs explain the length counterexample.
12	Transfer and explanation	Solve the constrained river-crossing task and discuss when it fails.
5	Exit evidence	Write and check the exit statement about displacement.

Question sequence

If I slide this arrow to another desk, what changes?
Can travelling farther leave you closer to your start?

Board development

Draw east 3, north 4 and the single diagonal; label distance 7 and displacement 5.
Display boat + current = ground velocity, then impose the zero horizontal component.

Expected correct and incorrect responses

A student gives −6 as a scaled length. Ask them to locate a negative distance on a ruler.
Correct: “The length is 6; the negative multiplier reverses the arrow.”
Incorrect: “The length is −6 because the multiplier is negative.”

Follow-up questions

If the current is faster than the boat, can a due-north path still exist? Explain with a squared component.

Remediation

Use two opposite paper arrows, then separately measure the two lengths and the zero resultant.

Exit check

State one quantity preserved by translating an arrow and one route property it does not encode.

Assessment checklist and sample responses

Credit the chosen orientation, signed components and a nonnegative magnitude separately. Sample: (−3,4)+(3,0)=(0,4), so crossing time is 20/4.
Correct: “The length is 6; the negative multiplier reverses the arrow.”