A vector-valued function gives position as a pair of components, each a function of time. Differentiate componentwise for velocity, and again for acceleration. Speed is the magnitude of velocity, never negative. Integrating speed gives distance; integrating velocity gives displacement.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Acceleration
Differentiate velocity: (2, 6t). At t = 1: (2, 6). The acceleration vector need not point along the velocity; here it is steeper, bending the path upward.
Direction of motion
The direction is the velocity vector’s direction: at t = 1, arctan(3/2) ≈ 56.3° above the x-axis. The unit tangent is (2, 3)/√13.
Distance traveled
Integrate the speed: ∫₀¹ √(4t² + 9t⁴) dt ≈ 1.44. That is the arc length of the path. On the calculator section, set it up and evaluate numerically.
Displacement differs
Integrate the velocity vector componentwise: ∫₀¹ (2t, 3t²) dt = (1, 1). The straight-line change in position, length √2 ≈ 1.41, less than the 1.44 traveled. After a closed loop the displacement is zero however far the particle went.
Integrating back
From acceleration (2, 6t) with v(0) = (1, 0): v = (2t + 1, 3t²). From that with r(0) = (0, 2): r = (t² + t, t³ + 2). Each step needs an initial condition to pin the constant in each component.
Read the question
"Distance" wants ∫ speed. "Displacement" or "position" wants ∫ velocity, componentwise. "How far from the start" wants the magnitude of the displacement. They differ by more than a sign.
Step 2: Try It Yourself
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Acceleration is (2, 6t), with v(0) = (1, 0) and r(0) = (0, 2). Find the velocity, the position, the speed at t = 1, and the displacement from t = 0 to 1
- Answerv = (2t + C₁, 3t² + C₂); v(0) = (1, 0) gives (2t + 1, 3t²)integrate componentwise, fix the constants
Step 0 of 4
- Vector a(3, 4) · length 5
Step 3: In Real Life
A satellite’s orbit
A satellite’s position is a vector function of time. Differentiate once for velocity, twice for acceleration, which always points at Earth. Mission control integrates the same vectors to predict where it will be.
Step 4: 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). At t = 1 that is (2, 3).
Step 5: 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?
5 of 8
Integrating speed gives what?
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 6: 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 traveled 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.
- Each integration back needs an initial condition per component.