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: Picture It
Start with something you already know.
A car’s dashboard
- The speedometer reads 30 miles per hour. It never says which way.
- The compass on the dashboard reads northeast. It never says how fast.
- Together they tell you how fast and which way the car is going.
- By evening the odometer has counted 40 miles since the car left home this morning.
- Yet the car is parked only 2 miles from home, in a straight line.
Velocity is that pair as one arrow, and its length is the speed, which is never negative.
The odometer adds up speed, so every mile counts: that is the distance traveled.
The straight line from home is the displacement, and a mile out and a mile back cancel in it.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A robot vacuum rolls 3 meters east, then turns and rolls 4 meters north. Its wheels have rolled 7 meters in all. How far is it now from where it started, in a straight line, in meters?
meters
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: Try It Yourself
Tap and try it out.
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 4: 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 5: Watch an Example
One step at a time.
Watch Maya Find Velocity, Speed and Acceleration
A particle has position r(t) = (3t, t² + 1) meters at time t seconds. Maya finds its velocity, speed and acceleration at t = 2, and says what each one means.
- Step 1
Maya differentiates each component on its own. The x-component 3t has derivative 3, and the y-component t² + 1 has derivative 2t, so the velocity is v(t) = (3, 2t) meters per second.
Watch Leo Work Back From Velocity, Then Finish It Yourself
A particle has velocity v(t) = (2t, 4) meters per second, and at t = 1 it is at (3, 5). Leo recovers its position and fixes the first constant. The second constant, the distance from the start, and one comparison are yours.
- Step 1
Integrating each component of v(t) = (2t, 4) on its own gives x = t² + C₁ and y = 4t + C₂. Each component gets its own constant, because a vector-valued function is two functions sharing one t.
Step 6: Your Turn
Practice makes it stick.
The Survey Drone
Problem 1 of 17
A survey drone flies with a constant velocity of (6, 8) meters per second for 5 seconds. How many meters does it fly?
meters
Paid by the Kilometer
Problem 2 of 17
A bike courier is paid $3 for every kilometer she rides. For the first half hour her velocity is (12, 5) kilometers per hour. For the next half hour it is (−12, −5), which brings her straight back to the shop. How much is she paid for the hour, in dollars?
dollars
Motion in the Plane
Each one is one or two moves: differentiate or integrate each component, then take a length when a speed or a straight-line distance is asked for.
Problem 3 of 17
A particle’s velocity is v = (−7, 24) meters per second. What is its speed, in meters per second?
m/s
Problem 4 of 17
A particle has position r(t) = (t³, 2t²) meters at time t seconds. What is its speed at t = 1, in meters per second?
m/s
Problem 5 of 17
A particle has position r(t) = (t³, 4t²) meters at time t seconds. What is the length of its acceleration vector at t = 1?
m/s²
Problem 6 of 17
A particle has velocity v(t) = (6t, 2t + 6) meters per second. How far is it at t = 2 from where it was at t = 0, in a straight line, in meters?
meters
Problem 7 of 17
A ball has acceleration a(t) = (0, −10) meters per second per second, and its velocity at t = 0 is v(0) = (5, 32) meters per second. What is its speed at t = 2, in meters per second?
m/s
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 17
Priya found the speed at t = 1 of a particle with position r(t) = (5 − 9t, 6t²) meters. One line is a slip. Which?
Priya’s speed at t = 1 for r(t) = (5 − 9t, 6t²)
- 1v(t) = (−9, 12t)
- 2v(1) = (−9, 12)
- 3(−9)² + 12² = −81 + 144 = 63
- Answerspeed = √63 ≈ 7.94
Problem 9 of 17
What is that particle’s speed at t = 1, in meters per second?
m/s
Problem 10 of 17
Ana was told a particle has acceleration a(t) = (6t, 4) meters per second per second and starting velocity v(0) = (2, 8) meters per second, and she found its speed at t = 1. One line is a slip. Which?
Ana’s speed at t = 1, from a(t) = (6t, 4) and v(0) = (2, 8)
- 1a(t) = (6t, 4)
- 2v(t) = (3t², 4t)
- 3v(1) = (3, 4)
- Answerspeed = √(9 + 16) = 5
Problem 11 of 17
With the constants fixed from v(0) = (2, 8), what is the particle’s speed at t = 1, in meters per second?
m/s
Problem 12 of 17
Sam was asked how far a particle traveled from t = 0 to t = 4, where v(t) = (6 − 3t, 8 − 4t) meters per second. One line is a slip. Which?
Sam’s distance traveled from t = 0 to t = 4, for v(t) = (6 − 3t, 8 − 4t)
- 1distance = ∫₀⁴ v(t) dt, one component at a time
- 2∫₀⁴ (6 − 3t) dt = 24 − 24 = 0
- 3∫₀⁴ (8 − 4t) dt = 32 − 32 = 0
- Answerdistance = 0 meters
Problem 13 of 17
How far did that particle travel from t = 0 to t = 4, in meters?
meters
Pick the Move
Choose the move or the reading that answers the question. Neither needs working all the way through.
Problem 14 of 17
A particle’s velocity is v(t) = (2t, 6) meters per second, and its position at t = 0 is (1, 4). Its position at t = 3 is wanted. Which move comes first?
Problem 15 of 17
At t = 2 a particle has velocity v(2) = (−3, 4) meters per second, and √((−3)² + 4²) = 5. Which sentence says what the 5 means?
Same Number?
Judge each one without a long calculation. Select every entry that fits.
Problem 16 of 17
In which of these trips is the distance traveled the same number as the length of the displacement?
Problem 17 of 17
Select every position below whose speed at t = 1 is 5.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A particle starts at the origin with velocity v(t) = (k − 2t, 4) meters per second, where k is a number. For which k is the particle back on the y-axis at t = 3?
Stretch 2 of 2
A particle starts at the origin with velocity v(t) = (6t, 8t) meters per second. At what time t, in seconds, has it traveled 80 meters?
seconds
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.
For the grown-up
The misconception this lesson targets is that distance traveled and displacement are the same number, so that integrating the velocity tells you how far something went. A student holding it gets every straight-line problem right and every return trip wrong: a courier who rides out and back integrates to zero kilometers, and a curved path is reported as its straight chord. Two smaller slips travel with it: treating a negative component as making the speed smaller (squaring −9 to −81), and dropping the constant in each component when working back from acceleration. The habit that settles all three is to read the question for its word first: "distance" adds up speed, which is never negative; "displacement" or "position" integrates velocity, one component and one constant at a time; and "how far from the start" is the length of that displacement.