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Chapter 12: Problem 3

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

Short Answer

Expert verified
Answer: The arc length of a curve can be expressed in terms of the speed of an object moving along the curve by using the arc length function: Arc_length(s) = 鈭玽(t) dt, where v(t) is the speed of the object at point t along the curve. This relationship depends on the specific function for the speed of the object.

Step by step solution

01

Define the arc length function of the curve

The arc length function of a curve represented by the position vector r(t) is given by the integral of the magnitude of the derivative of r(t) with respect to t, which is also called the tangent vector T(t) at point t. Here's the arc length function: Arc_length(s) = 鈭珅|T(t)|| dt
02

Compute the tangent vector T(t)

The tangent vector T(t) is given by the derivative of r(t) with respect to t: T(t) = dr(t)/dt
03

Calculate the magnitude of the tangent vector T(t)

The magnitude of the tangent vector T(t) represents the speed of the object moving along the curve at point t. It is given by: ||T(t)|| = sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) where x(t), y(t), and z(t) are the components of the position vector r(t).
04

Express arc length in terms of the speed

Since the speed v(t) of an object moving along the curve is the magnitude of the tangent vector T(t), we can substitute v(t) into our arc length function: Arc_length(s) = 鈭玽(t) dt Here, the arc length is expressed in terms of the speed v(t) of an object moving along the curve. The exact relationship would depend on the specific function for the speed of the object.

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