Vectors in context: velocity and force

About this lesson

Algebra II › Vectors and matrices

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Vectors in context: velocity and force

Warm-up

Three quick questions from earlier lessons.

Discover

A swimmer swims north at 3 km/h across a river flowing east at 4 km/h. How fast does she move over the ground?

  1. 5 km/h
  2. 7 km/h

Key idea

Velocities and forces are vectors: when several act together, add them.

The resultant of several vectors is their sum. Its magnitude is the speed (or total force), and its direction angle is where the object actually goes. Write each vector in components first.

Worked example

Forces of 30 N due east and 40 N due north pull on a ring. Find the resultant.

F⃗=⟨30,40⟩\vec{F} = \langle 30, 40 \rangle
Components.
∣F⃗∣=50|\vec{F}| = 50 N
√(900 + 1600).
θ=tan⁡−143≈53.1∘\theta = \tan^{-1}\dfrac{4}{3} \approx 53.1^\circ north of east
tan θ = 40/30.

Your turn, guided

Your turn: a plane flies east at 200 mph with a wind from the south at 50 mph.

Ground velocity ⟨200,50⟩\langle 200, 50 \rangle
East and north components.
Speed =42500≈206.2= \sqrt{42500} \approx 206.2 mph
√(40000 + 2500) ≈ 206.2: type the exact value.

Practice ★

  1. A plane flies due east at 101 miles per hour through the air, and the wind blows due north at 32 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  2. A plane flies due east at 236 miles per hour through the air, and the wind blows due north at 60 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  3. A plane flies due east at 174 miles per hour through the air, and the wind blows due north at 17 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale

Watch out!

Practice ★★

  1. Two people pull a box: one with 36 newtons due east, the other with 59 newtons due north. At what angle north of east does the box start to move? Give degrees to 1 decimal place.Not to scale
  2. Two people pull a box: one with 27 newtons due east, the other with 23 newtons due north. At what angle north of east does the box start to move? Give degrees to 1 decimal place.Not to scale
  3. Two people pull a box: one with 88 newtons due east, the other with 72 newtons due north. At what angle north of east does the box start to move? Give degrees to 1 decimal place.Not to scale

Exit ticket

Three questions to check you have got it.

  1. A plane flies due east at 101 miles per hour through the air, and the wind blows due north at 32 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  2. A plane flies due east at 236 miles per hour through the air, and the wind blows due north at 60 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  3. A plane flies due east at 174 miles per hour through the air, and the wind blows due north at 17 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale

Summary card

Vectors in context

  1. Write each velocity or force in components.
  2. Add them: the resultant.
  3. Magnitude = speed or total force; tan⁡θ\tan\theta = direction.

Watch out: speeds in different directions do not simply add.

End of the lesson

Next lesson: Matrices: representing data, adding and multiplying by a scalar Back to the chapter

Answers: Vectors in context: velocity and force

A

  1. 54495 \sqrt{449}
  2. 437064 \sqrt{3706}
  3. 30565\sqrt{30565}

B

  1. 180tan⁡−1(5936)π\frac{180 \tan^{-1}{\left(\frac{59}{36} \right)}}{\pi}
  2. 180tan⁡−1(2327)π\frac{180 \tan^{-1}{\left(\frac{23}{27} \right)}}{\pi}
  3. 180tan⁡−1(911)π\frac{180 \tan^{-1}{\left(\frac{9}{11} \right)}}{\pi}

C

  1. 54495 \sqrt{449}
  2. 437064 \sqrt{3706}
  3. 30565\sqrt{30565}

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