To differentiate r(t), differentiate each component separately. Nothing new is required beyond single-variable calculus.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Velocity
r′(t) is the velocity vector. It points along the direction of travel and is tangent to the curve.
Speed is a number
Speed is the magnitude of velocity. Velocity says how fast and which way; speed drops the direction.
Acceleration
r″(t) is acceleration. It need not point along the curve, and usually does not.
Circular motion
On a circle at constant speed the acceleration points straight at the centre. Speed is unchanged while direction changes constantly.
The unit tangent
Dividing velocity by speed gives the unit tangent vector T. It records the direction of travel with the speed stripped out.
Velocity is the derivative vector
Differentiating r(t) componentwise gives velocity, which is tangent to the curve. Its magnitude is speed, a scalar, and the distinction between them is used constantly.
Acceleration has two effects
Differentiating velocity gives acceleration. Part of it changes speed and part changes direction, which is why circular motion at constant speed still has nonzero acceleration.
Constant speed means perpendicular acceleration
If the speed never changes, the acceleration is always perpendicular to the velocity. It follows from differentiating v · v and is the reason centripetal acceleration points inwards.
Integrating recovers position
Integrating acceleration gives velocity up to a constant vector, and again gives position. Two initial conditions fix both constants, exactly as in one-dimensional motion.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 4)
- Slope of the tangent4
Step 3: Watch an Example
One step at a time.
Watch Elena Find a Speed
Elena has r(t) = ⟨3t, 4t, 0⟩ and needs the speed.
- Step 1
She differentiates each component to get r′(t) = ⟨3, 4, 0⟩.
Step 4: Your Turn
Practice makes it stick.
The Drone Path
Problem 1 of 2
r(t) = ⟨t², 6t, 0⟩. What is the x-component of velocity at t = 5?
The Straight Run
Problem 2 of 2
r(t) = ⟨6t, 8t, 0⟩. What is the speed?
Differentiate the Motion
1 of 8
r(t) = ⟨t³, t, 0⟩. What is the x-component of velocity at t = 2?
2 of 8
r(t) = ⟨t³, t, 0⟩. What is the x-component of acceleration at t = 2?
3 of 8
r(t) = ⟨5t, 0, 12t⟩. What is the speed?
4 of 8
Velocity is ⟨0, 0, 0⟩ at some t. Has the particle stopped? 1 yes, 0 no.
5 of 8
A velocity vector of magnitude 9. What is the magnitude of the unit tangent?
6 of 8
r(t) = ⟨7t, −4, 0⟩. What is the y-component of velocity?
7 of 8
Match each quantity to its description.
Tap a card on the left to start.
8 of 8
A particle on a circle at constant speed. Is its acceleration zero? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
r(t) = ⟨t², 2t, 0⟩. What is the y-component of velocity?
Question 2 of 2
How do velocity and speed differ?
What You Learned
- Differentiate a vector-valued function one component at a time.
- Velocity is r′ and is tangent to the curve; speed is its magnitude.
- Acceleration is r″ and generally does not point along the curve.