Velocity, Acceleration, and Arc Length
students, in this lesson you will learn how a moving object is described with vector-valued functions, how to find its velocity and acceleration, and how to measure the distance it travels along a curved path 🚗✈️. These ideas connect calculus with real motion in space, such as a drone flying, a planet orbiting, or a runner moving on a track.
What you will learn
By the end of this lesson, you should be able to:
- Explain what $\mathbf{r}(t)$, $\mathbf{v}(t)$, and $\mathbf{a}(t)$ mean.
- Compute velocity and acceleration from a position function.
- Find speed and understand the difference between speed and velocity.
- Use the arc length formula to measure the length of a curve.
- Connect these ideas to motion along a line or a path in space.
The big idea is this: a vector-valued function tells where an object is at each time $t$. From that position, calculus lets us study how it moves and how far it travels.
Position, velocity, and acceleration
A moving particle in space is often described by a position vector function
$$\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle.$$
This means that at time $t$, the object is at the point $(x(t), y(t), z(t))$. If the motion happens in a plane, we may use
$$\mathbf{r}(t) = \langle x(t), y(t) \rangle.$$
The velocity vector is the derivative of position:
$$\mathbf{v}(t) = \mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle.$$
Velocity tells both how fast and in what direction the object is moving. The direction matters because a vector has direction as well as size.
The acceleration vector is the derivative of velocity:
$$\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t) = \langle x''(t), y''(t), z''(t) \rangle.$$
Acceleration measures how velocity changes over time. That change can happen because speed changes, direction changes, or both. For example, a car turning a corner has acceleration even if its speed stays constant, because its direction changes.
A simple example
Suppose
$$\mathbf{r}(t) = \langle t, t^2 \rangle.$$
Then
$$\mathbf{v}(t) = \langle 1, 2t \rangle$$
and
$$\mathbf{a}(t) = \langle 0, 2 \rangle.$$
At $t=1$, the velocity is
$$\mathbf{v}(1) = \langle 1, 2 \rangle,$$
so the object is moving right and upward. The acceleration is constant:
$$\mathbf{a}(1) = \langle 0, 2 \rangle.$$
This means the upward part of the motion is getting stronger over time.
Speed is not the same as velocity
The speed of the object is the magnitude of the velocity vector:
$$|\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}.$$
In two dimensions, this becomes
$$|\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}.$$
Speed is always nonnegative because it is a length. Velocity, however, is a vector, so it can point in many directions.
This distinction is important in real life. If a runner circles a track at a steady pace, the speed may stay the same, but the velocity keeps changing because the direction is changing the whole time 🏃♀️.
Example: motion with constant speed
Let
$$\mathbf{r}(t) = \langle \cos t, \sin t \rangle.$$
Then
$$\mathbf{v}(t) = \langle -\sin t, \cos t \rangle.$$
The speed is
$$|\mathbf{v}(t)| = \sqrt{(-\sin t)^2 + (\cos t)^2} = \sqrt{\sin^2 t + \cos^2 t} = 1.$$
So the object moves around the unit circle with constant speed $1$. Even though the speed is constant, the velocity changes direction at every moment.
Acceleration and changing motion
Acceleration describes how velocity changes. There are two common ways this happens:
- The speed changes.
- The direction changes.
If an object moves in a straight line but speeds up, acceleration points along the line of motion. If an object moves around a curve, acceleration can point inward toward the center of the curve.
For example, suppose
$$\mathbf{r}(t) = \langle t, t^2, t^3 \rangle.$$
Then
$$\mathbf{v}(t) = \langle 1, 2t, 3t^2 \rangle$$
and
$$\mathbf{a}(t) = \langle 0, 2, 6t \rangle.$$
At $t=2$,
$$\mathbf{v}(2) = \langle 1, 4, 12 \rangle$$
and
$$\mathbf{a}(2) = \langle 0, 2, 12 \rangle.$$
This tells us the object is moving in a 3D direction that is changing over time, with acceleration affecting the path in more than one coordinate direction.
Arc length: how far the curve is traveled
The arc length of a curve gives the actual distance traveled along the path, not just the straight-line distance between start and end points. This is especially useful when motion is curved.
If a particle moves along the vector-valued function
$$\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$$
for $a \le t \le b$, then the arc length is
$$L = \int_a^b |\mathbf{r}'(t)|\,dt.$$
Since $\mathbf{r}'(t) = \mathbf{v}(t)$, this can also be written as
$$L = \int_a^b |\mathbf{v}(t)|\,dt.$$
In words, arc length is the integral of speed over time. That makes sense physically: distance traveled equals how fast you move, added up over the time interval ⏱️.
Why arc length matters
Imagine a delivery drone flying from one warehouse to another. The straight-line distance between the two points may be short, but the drone might follow a curved path to avoid buildings or no-fly zones. The arc length is the true flight distance.
Example: length of a simple curve
Let
$$\mathbf{r}(t) = \langle 3t, 4t \rangle, \quad 0 \le t \le 2.$$
First find the velocity:
$$\mathbf{r}'(t) = \langle 3, 4 \rangle.$$
Then the speed is
$$|\mathbf{r}'(t)| = \sqrt{3^2 + 4^2} = 5.$$
So the arc length is
$$L = \int_0^2 5\,dt = 10.$$
This matches the fact that the motion is along a straight line with constant speed.
Example: curved motion
Let
$$\mathbf{r}(t) = \langle \cos t, \sin t \rangle, \quad 0 \le t \le \pi.$$
Then
$$\mathbf{r}'(t) = \langle -\sin t, \cos t \rangle,$$
so
$$|\mathbf{r}'(t)| = 1.$$
Therefore,
$$L = \int_0^\pi 1\,dt = \pi.$$
This is the length of the upper half of the unit circle.
Connecting these ideas to lines, planes, and vector-valued functions
Velocity, acceleration, and arc length all come from the study of vector-valued functions. A line in space can be written in vector form, and a moving object can also be described by a vector function. In both cases, vectors describe location and direction.
A line may be written as
$$\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v},$$
where $\mathbf{r}_0$ is a point on the line and $\mathbf{v}$ is a direction vector. A moving particle uses the same style of notation, but the vector now changes over time. The path of the particle is the curve traced out by $\mathbf{r}(t)$.
Planes also use vectors. A plane can be described by a point and a normal vector. While the plane itself does not involve motion, the same vector ideas help us understand how a particle may move relative to that plane or how a curve may intersect it.
This is why velocity, acceleration, and arc length belong in the unit on lines, planes, and vector-valued functions: they are applications of vectors and derivatives in space.
How to solve common problems
When you are given a position function, follow these steps:
- Find velocity by differentiating $\mathbf{r}(t)$.
- Find acceleration by differentiating $\mathbf{v}(t)$.
- Find speed using the magnitude $|\mathbf{v}(t)|$.
- Find arc length by integrating speed: $$L = \int_a^b |\mathbf{r}'(t)|\,dt.$$
A good habit is to check whether the answer makes sense. For example, arc length should be a positive number, and speed should never be negative.
Conclusion
students, velocity, acceleration, and arc length give calculus a way to describe real motion in space. Velocity tells where an object is headed, acceleration tells how the motion changes, and arc length tells how far the object actually travels. Together, these ideas show how vector-valued functions turn geometry into motion and motion into calculus. They are central tools for understanding paths, curves, and movement in Multivariable Calculus ✨.
Study Notes
- A position function is written as $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ or $\mathbf{r}(t) = \langle x(t), y(t) \rangle$.
- Velocity is the derivative of position: $\mathbf{v}(t) = \mathbf{r}'(t)$.
- Acceleration is the derivative of velocity: $\mathbf{a}(t) = \mathbf{r}''(t)$.
- Speed is the magnitude of velocity: $|\mathbf{v}(t)|$.
- Speed is a scalar, while velocity and acceleration are vectors.
- Arc length over $a \le t \le b$ is $$L = \int_a^b |\mathbf{r}'(t)|\,dt.$$
- Arc length equals the integral of speed over time.
- A curve can be traced by a vector-valued function, connecting motion to geometry.
- Constant speed does not mean constant velocity, because direction can change.
- These ideas are foundational for describing motion in space, especially in the study of lines, planes, and vector-valued functions.