Question

Express the arc length of a curve in terms of the speed of an object moving along the curve.

Short answer

Question: Express the speed of an object moving along a curve in terms of the arc length of the curve. Answer: The speed (v) of an object moving along a curve can be expressed in terms of the arc length as follows: v = √((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2)

Step-by-step solution

Write the arc length formula

For a curve represented as a function of a parameter t, the arc length (s) can be found using the following formula: s = Integral[||d**r|| dt] Here, r(t) is a position vector of a point on the curve, ||d**r|| is the magnitude of the curve's tangent vector, and t is the parameter.

Find the tangent vector

The tangent vector at a point on the curve is given by the derivative of the position vector of that point with respect to the parameter t. Tangent vector = d**r/dt

Find the magnitude of the tangent vector

To find the magnitude (length) of the tangent vector, take the square root of the sum of the squares of its components, i.e., ||d**r|| = √((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) Here, x(t), y(t), z(t) are the components of the position vector r(t).

Write arc length in terms of magnitude of the tangent vector

Now, substitute the expression for the magnitude of the tangent vector into the arc length formula: s = Integral[√((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2)] dt

Differentiate the arc length with respect to time

To express the arc length in terms of the speed, differentiate the arc length (s) with respect to time (t): ds/dt = d[√((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2)] dt The derivative of the arc length with respect to time (ds/dt) is the speed (v) of the object moving along the curve. Therefore, we have: Speed (v) = √((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) This gives us the expression for the speed of an object moving along the curve in terms of the arc length.