A vector-valued function gives position as a pair of components, each a function of time.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Differentiate componentwise
The velocity vector differentiates each component separately. Acceleration differentiates again.
Speed is a scalar
Velocity is a vector; speed is its magnitude, √(x′² + y′²). Speed has no direction and is never negative.
Distance travelled
Integrating the speed over an interval gives the distance travelled, which is the arc length of the path.
Displacement differs
Integrating the velocity vector gives displacement, which can be zero after a closed loop even when distance is large.
Integrating back
Integrating acceleration gives velocity, and integrating velocity gives position. Each step needs an initial condition to pin the constant.
Position as a vector function
r(t) = ⟨x(t), y(t)⟩ packages both coordinates into one object. Differentiating componentwise gives velocity, and differentiating again gives acceleration. Vector notation compresses the whole motion.
Velocity, speed, acceleration
Velocity is a vector, speed its magnitude, acceleration the derivative of velocity. A particle can have constant speed and nonzero acceleration if it is turning, which is a favourite exam point.
Displacement against distance
Integrating the velocity vector gives displacement, a vector. Integrating speed gives total distance travelled, a scalar. Confusing them is the most common error in vector motion problems.
Recovering position from velocity
Integrating velocity componentwise recovers position up to a constant vector, which an initial position fixes. The structure is identical to one-dimensional motion, applied twice.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 4) · length 5
Step 3: Watch an Example
One step at a time.
Watch Diego Find Speed From Position
A particle has position (t², t³) and Diego needs its speed at t = 1.
- Step 1
Differentiating each component gives velocity (2t, 3t²).
Step 4: Your Turn
Practice makes it stick.
The Speed
Problem 1 of 2
Velocity components 3 and 4. What is the speed?
The Velocity
Problem 2 of 2
Position (t², t³) gives velocity (2t, 3t²). What is the x component at t = 3?
Motion in the Plane
1 of 8
Velocity components 6 and 8. Speed?
2 of 8
Velocity components 5 and 12. Speed?
3 of 8
Position (t², t³) gives velocity (2t, 3t²). y component at t = 2?
4 of 8
Can speed be negative? 1 yes, 0 no.
5 of 8
Integrating speed gives what? 1 distance travelled, 2 displacement.
6 of 8
A particle returns to its start. What is its displacement magnitude?
7 of 8
Sort each quantity by whether it is a vector.
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- Empty
- Empty
8 of 8
Velocity components 8 and 15. Speed?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Velocity components 9 and 12. What is the speed?
Question 2 of 2
What is the difference between distance travelled and displacement?
What You Learned
- Differentiate a vector-valued function componentwise.
- Speed is the magnitude of velocity and is never negative.
- Integrating speed gives distance; integrating velocity gives displacement.