Connecting the Motion Quantities
A parametric-motion problem often asks for several related quantities at once. The position tells where an object is; velocity describes how its position changes; acceleration describes how velocity changes; speed measures the magnitude of velocity; and arc length gives the distance traveled along the path.
These quantities are connected, but they are not interchangeable. In particular, displacement is a change in position, while distance traveled is the length of the path between two parameter values. A reliable solution keeps track of the parameter interval and checks what each calculated quantity means.
If the curve is continuously differentiable on \([a,b]\), its arc length over that parameter interval is the integral of its speed. For a piecewise continuously differentiable curve, calculate the length on each smooth piece and add the results. A corner or a change in velocity at a join does not, by itself, prevent this piece-by-piece calculation.
Displacement is instead the difference between the final and initial position vectors, \(\mathbf{r}(b)-\mathbf{r}(a)\). Its magnitude is the straight-line distance between the endpoints, not necessarily the distance traveled along the curve.
A Useful Test for Speeding Up
The velocity–acceleration dot product gives a concise test for whether speed is increasing or decreasing. The test requires nonzero velocity at the instant being considered: at a point where speed is zero, the unit direction of motion is not defined, and the speed derivative formula below cannot be used.
To justify the test, write speed as the square root of the dot product of velocity with itself. Differentiate using the chain rule and the product rule:
The denominator is positive when velocity is nonzero, so the sign of the speed derivative is exactly the sign of \(\mathbf{v}\cdot\mathbf{a}\). This explains why acceleration alone does not decide whether an object is speeding up: its component in the direction of velocity matters.
Worked Example: Motion Around a Quarter-Circle
Worked Example: Motion Around a Quarter-Circle
An object moves according to \(\mathbf{r}(t)=\langle 3\cos t,3\sin t\rangle\), for \(0\leq t\leq \frac{\pi}{2}\). Coordinates are in meters, and \(t\) is in seconds with angular rate \(1\) radian per second. Find velocity, acceleration, speed, direction at \(t=\frac{\pi}{3}\), whether speed is changing, arc length, and displacement.
Differentiate each coordinate:
The speed is constant because
At \(t=\frac{\pi}{3}\), use \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\) and \(\cos\frac{\pi}{3}=\frac12\). Then
To check whether the object is speeding up, calculate the dot product for any \(t\):
Since speed is \(3\) throughout, the zero dot product agrees with the conclusion that speed is constant. The arc length on the stated interval is
As a check, differentiating the antiderivative \(3t\) gives the speed integrand \(3\). The endpoints are \(\mathbf{r}(0)=\langle 3,0\rangle\) and \(\mathbf{r}(\frac{\pi}{2})=\langle 0,3\rangle\), so the displacement is \(\langle-3,3\rangle\) m, with magnitude \(3\sqrt{2}\) m. That is less than the arc length, as expected for a curved path.
Worked Example: Speeding Up Along a Polynomial Curve
Worked Example: Speeding Up Along a Polynomial Curve
Let \(\mathbf{r}(t)=\langle t^2,\frac{2}{3}t^3\rangle\) for \(0\leq t\leq1\). Treat the coordinate values as meters and \(t\) as seconds, with coefficients carrying the units needed for these formulas. Find the velocity, acceleration, speed, direction and speed change at \(t=1\), and total arc length.
Differentiate to obtain
Therefore, the speed is \(\sqrt{4t^2+4t^4}=2t\sqrt{1+t^2}\), since \(t\geq0\) on the interval. At \(t=1\), the velocity is \(\langle2,2\rangle\) m/s and the speed is \(2\sqrt{2}\) m/s. The direction unit vector is \(\langle1/\sqrt{2},1/\sqrt{2}\rangle\).
At that instant the acceleration is \(\langle2,4\rangle\) m/s², and
Thus the object is speeding up at \(t=1\). In fact, for the whole interval, \(\mathbf{v}(t)\cdot\mathbf{a}(t)=4t+8t^3\), which is positive for \(t>0\). At \(t=0\), velocity is zero, so the nonzero-velocity test does not apply there.
For arc length, integrate the speed. With \(u=1+t^2\), \(du=2t\,dt\):
Differentiate \(\frac{2}{3}(1+t^2)^{3/2}\) to check the integration: the result is \(\frac{2}{3}\cdot\frac32(1+t^2)^{1/2}\cdot2t=2t\sqrt{1+t^2}\), the original speed. The endpoint positions are \(\langle0,0\rangle\) and \(\langle1,\frac23\rangle\) m, so displacement has magnitude \(\sqrt{1+\frac49}=\frac{\sqrt{13}}{3}\) m. It is a straight-line endpoint distance, not the arc length.
Worked Example: Adding Length Across a Piecewise Path
Worked Example: Adding Length Across a Piecewise Path
An object follows this position function, with coordinates in meters and \(t\) in seconds:
Find the total distance traveled, describe the direction and speed on each piece, and decide whether velocity exists at the join \(t=1\).
On the first piece, \(\mathbf{v}(t)=\langle6t,8t\rangle\), so speed is \(\sqrt{36t^2+64t^2}=10t\) because \(t\geq0\). For \(t>0\), the direction is the constant unit vector \(\langle\frac35,\frac45\rangle\). Its length is
On the second piece, \(\mathbf{v}(t)=\langle2,0\rangle\), so the speed is \(2\) m/s and the direction is \(\langle1,0\rangle\). The length over \(1\leq t\leq3\) is
Thus total distance traveled is \(L_1+L_2=5+4=9\) m. The position agrees from both formulas at the join: \(\mathbf{r}(1)=\langle3,4\rangle\) m. But the one-sided velocities differ: the first-piece velocity at \(1\) is \(\langle6,8\rangle\) m/s, and the second-piece velocity is \(\langle2,0\rangle\) m/s. Therefore velocity is not defined at the join. The path has a corner there, yet its length is still the sum of the two piece lengths.
As a final comparison, the object starts at \(\langle0,0\rangle\) and ends at \(\langle7,4\rangle\) m. Its displacement magnitude is \(\sqrt{7^2+4^2}=\sqrt{65}\) m, which is not the \(9\) m distance traveled along the path.
A Reliable Workflow and a Common Pitfall
For a problem combining motion and length, make the parameter interval explicit before calculating. Then differentiate coordinate functions, form velocity and acceleration, and use the velocity norm for speed. Find direction only where speed is nonzero. For length, integrate speed over the requested interval, splitting at any piecewise joins.
State the parameter values being considered and track whether coordinates, velocity, acceleration, and length are measured in meters, meters per second, meters per second squared, and meters.
Differentiate position to obtain velocity and acceleration; take the magnitude of velocity for speed.
Use the dot product only when velocity is nonzero, integrate speed for distance traveled, and compare displacement with path length when both are available.
A frequent error is to report displacement magnitude as distance traveled. For the quarter-circle example, a wrong calculation would be \(\sqrt{(-3)^2+3^2}=3\sqrt{2}\) m and then call that the distance traveled. That calculation gives the straight-line separation of the endpoints. The correction is to integrate speed over the interval: \(\int_0^{\pi/2}3\,dt=\frac{3\pi}{2}\) m. The distinction matters whenever the path is not a straight segment.
Also avoid using acceleration magnitude as speed. Acceleration is the rate of change of velocity; speed is the magnitude of velocity. To determine whether speed is increasing, check \(\mathbf{v}\cdot\mathbf{a}\) where \(\mathbf{v}\ne\mathbf{0}\), not just whether the acceleration vector is large.
Check Your Understanding
Use the parameter interval stated in each question, and include units when quantities represent physical measurements.
- For \(\mathbf{r}(t)=\langle 2\cos t,2\sin t\rangle\) on \(0\leq t\leq\pi\), find speed and arc length. How does the displacement magnitude compare with the length?
- If \(\mathbf{v}(t)=\langle t,2\rangle\) and \(\mathbf{a}(t)=\langle1,0\rangle\), is the object speeding up or slowing down at \(t=3\)? Show the dot-product test.
- Why is \(\mathbf{v}/\|\mathbf{v}\|\) not defined when \(\mathbf{v}=\mathbf{0}\)? What does this mean for interpreting direction at that instant?
- A path consists of two smooth pieces that meet at a corner. Can total arc length still be found by adding the lengths of the pieces? Explain.
- In one sentence, distinguish displacement from distance traveled.
