Vector-valued functions: velocity, speed, acceleration

About this lesson

Calculus BC (AP-aligned) › Parametric equations, polar coordinates and vector-valued functions

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Vector-valued functions: velocity, speed, acceleration

Warm-up

Three quick questions from earlier lessons.

Discover

r⃗(t)=⟨t2,3t⟩\vec r(t) = \langle t^2, 3t \rangle gives a position at each time. Its velocity is v⃗(t)=⟨2t,3⟩\vec v(t) = \langle 2t, 3 \rangle. At t=2t = 2, v⃗=⟨4,3⟩\vec v = \langle 4, 3 \rangle. What is the speed?

  1. 5
  2. 7

Key idea

Differentiate and integrate componentwise.

r⃗(t)=⟨x(t),y(t)⟩\vec r(t) = \langle x(t), y(t) \rangle: velocity v⃗=r⃗ ′\vec v = \vec r\,', acceleration a⃗=v⃗ ′\vec a = \vec v\,', speed ∣v⃗∣=x′2+y′2|\vec v| = \sqrt{x'^2 + y'^2}. Position from velocity: r⃗(t)=r⃗(0)+∫0tv⃗(s) ds\vec r(t) = \vec r(0) + \displaystyle\int_0^t \vec v(s)\,ds.

Worked example

r⃗(t)=⟨cos⁡t,sin⁡t⟩\vec r(t) = \langle \cos t, \sin t \rangle.

v⃗(t)=⟨−sin⁡t,cos⁡t⟩\vec v(t) = \langle -\sin t, \cos t \rangle
Componentwise.
Speed =1= 1
sin² + cos² = 1.
a⃗(t)=⟨−cos⁡t,−sin⁡t⟩\vec a(t) = \langle -\cos t, -\sin t \rangle
Points towards the center.

Your turn, guided

Your turn: v⃗(t)=⟨3t2,2⟩\vec v(t) = \langle 3t^2, 2 \rangle, r⃗(0)=⟨0,5⟩\vec r(0) = \langle 0, 5 \rangle. Find y(1)y(1).

y(1)=7y(1) = 7
5 + ∫₀¹ 2 dt.

Practice ★

  1. A particle moves with position r⃗(t)=⟨t2,6t⟩\vec r(t) = \langle t^2, 6t \rangle. Find its speed at t=2t = 2. Give an exact value.
  2. A particle moves with position r⃗(t)=⟨t2,5t⟩\vec r(t) = \langle t^2, 5t \rangle. Find its speed at t=4t = 4. Give an exact value.
  3. A particle moves with position r⃗(t)=⟨t2,3t⟩\vec r(t) = \langle t^2, 3t \rangle. Find its speed at t=1t = 1. Give an exact value.

Watch out!

Practice ★★

  1. r⃗(t)=⟨6t2,8sin⁡t⟩\vec r(t) = \langle 6t^2, 8\sin t \rangle. Find the magnitude of the acceleration at t=0t = 0.
  2. r⃗(t)=⟨8t2,5sin⁡t⟩\vec r(t) = \langle 8t^2, 5\sin t \rangle. Find the magnitude of the acceleration at t=0t = 0.
  1. v⃗(t)=⟨2t,3⟩\vec v(t) = \langle 2t, 3 \rangle and r⃗(0)=⟨1,0⟩\vec r(0) = \langle 1, 0 \rangle. Find the xx-coordinate of the position at t=2t = 2.
  2. v⃗(t)=⟨2t,6⟩\vec v(t) = \langle 2t, 6 \rangle and r⃗(0)=⟨1,0⟩\vec r(0) = \langle 1, 0 \rangle. Find the xx-coordinate of the position at t=5t = 5.

Exit ticket

Three questions to check you have got it.

  1. A particle moves with position r⃗(t)=⟨t2,6t⟩\vec r(t) = \langle t^2, 6t \rangle. Find its speed at t=2t = 2. Give an exact value.
  2. A particle moves with position r⃗(t)=⟨t2,5t⟩\vec r(t) = \langle t^2, 5t \rangle. Find its speed at t=4t = 4. Give an exact value.
  3. A particle moves with position r⃗(t)=⟨t2,3t⟩\vec r(t) = \langle t^2, 3t \rangle. Find its speed at t=1t = 1. Give an exact value.

Summary card

Vector motion

  • v⃗=r⃗ ′\vec v = \vec r\,', a⃗=v⃗ ′\vec a = \vec v\,', speed =∣v⃗∣= |\vec v|.
  • Position: start + ∫v⃗\int \vec v.

Watch out: speed is a length, not a sum of components.

End of the lesson

Next lesson: Slopes of polar curves Back to the chapter

Answers: Vector-valued functions: velocity, speed, acceleration

A

  1. 2132 \sqrt{13}
  2. 89\sqrt{89}
  3. 13\sqrt{13}

B

  1. 1212
  2. 1616

C

  1. 55
  2. 2626

D

  1. 2132 \sqrt{13}
  2. 89\sqrt{89}
  3. 13\sqrt{13}

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