Motion along a space curve describes an object whose position changes in three dimensions, such as a drone flying through air or a planet moving through space. Calculus gives a precise way to describe the path using a vector-valued position function r(t). From this function, we can find velocity, speed, acceleration, and the total distance traveled.
This matters because many real motions are not straight lines and must be analyzed using both algebra and geometry.
The derivative r'(t) gives the velocity vector, which points in the direction of motion and has magnitude equal to speed. The arc length function measures distance along the curve, making it possible to reparameterize motion by distance instead of time. When a curve is described by arc length s, the tangent vector has unit length, which simplifies geometric analysis.
Acceleration can be split into tangential and normal parts to show how speed changes and how the path bends.
Key Facts
- Position along a space curve is written r(t) = <x(t), y(t), z(t)>.
- Velocity is v(t) = r'(t) = <x'(t), y'(t), z'(t)>.
- Speed is |v(t)| = |r'(t)| = sqrt((x'(t))^2 + (y'(t))^2 + (z'(t))^2).
- Acceleration is a(t) = v'(t) = r''(t).
- Arc length from t = a to t = b is L = integral from a to b of |r'(t)| dt.
- For arc-length parameter s, the unit tangent is T(s) = dr/ds and |T(s)| = 1.
Vocabulary
- Space curve
- A space curve is a path in three-dimensional space described by a vector-valued function.
- Position vector
- A position vector r(t) gives the location of a moving particle at time t.
- Velocity vector
- A velocity vector is the derivative of position and gives both the direction and rate of motion.
- Arc length
- Arc length is the distance measured along a curve between two parameter values.
- Unit tangent vector
- A unit tangent vector points in the direction of motion along the curve and has length 1.
Common Mistakes to Avoid
- Confusing speed with velocity. Speed is the scalar magnitude |v(t)|, while velocity is a vector with direction.
- Using straight-line distance instead of arc length. The distance traveled along a curved path is found by integrating speed, not by subtracting endpoints.
- Forgetting to differentiate each component of r(t). Velocity and acceleration require taking the derivative of x(t), y(t), and z(t) separately.
- Assuming the parameter t is always arc length. A curve is parameterized by arc length only when |r'(t)| = 1 or when the parameter has been changed to s.
Practice Questions
- 1 For r(t) = <t, t^2, 2t> on 0 <= t <= 2, find v(t), a(t), and the speed at t = 1.
- 2 For r(t) = <3cos t, 3sin t, 4t>, find the speed and the arc length from t = 0 to t = pi.
- 3 A particle moves along the same geometric curve twice, once with constant speed and once with changing speed. Explain which quantities depend on the path only and which depend on how the particle moves along the path.