Velocity and Acceleration

Differentiate each component and get motion.

12 problems · about 23 minutes · N-VM.A.1

What this lesson teaches

The student differentiates vector-valued functions to find velocity, speed and acceleration.

Straightforward practice. Get the method working first.

5 problems

r(t) = ⟨t², 2t, 0⟩. What is the y-component of velocity?
Answer 2
Why 2.
How do velocity and speed differ?
Answer Velocity is a vector; speed is its magnitude.
Why Speed is velocity with the direction removed.
r(t) = ⟨t³, t, 0⟩. What is the x-component of velocity at t = 2?
Answer 12
Why 3t² at t = 2.
r(t) = ⟨t³, t, 0⟩. What is the x-component of acceleration at t = 2?
Answer 12
Why 6t at t = 2.
r(t) = ⟨5t, 0, 12t⟩. What is the speed?
Answer 13
Why 25 + 144 = 169.

The same ideas with more to keep track of.

3 problems

Velocity is ⟨0, 0, 0⟩ at some t. Has the particle stopped? 1 yes, 0 no.
Answer 1
Why Zero velocity is no motion.
A velocity vector of magnitude 9. What is the magnitude of the unit tangent?
Answer 1
Why Dividing by the speed normalises it.
r(t) = ⟨7t, −4, 0⟩. What is the y-component of velocity?
Answer 0
Why A constant differentiates to zero.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each quantity to its description.
Answer Velocity → The first derivative, a vector; Speed → The magnitude of the first derivative, a number; Acceleration → The second derivative, a vector
Why One of the three is not a vector.
A particle on a circle at constant speed. Is its acceleration zero? 1 yes, 0 no.
Answer 0
Why The direction keeps changing.
The Drone Path: r(t) = ⟨t², 6t, 0⟩. What is the x-component of velocity at t = 5?
Answer 10
Why 2 × 5 = 10.
The Straight Run: r(t) = ⟨6t, 8t, 0⟩. What is the speed?
Answer 10
Why 10.