Mathematical Tools for Physics

13—Vector Calculus 2 398

Ifk→ 0 does this give the correct answer?

Weighted Integrals
The time for a particle to travel along a short segment of a path isdt=ds/vwherevis the speed. The total
time along a path is of course the integral ofdt.

T=

∫

dt=

∫

ds

v

How much time does it take a particle to slide down a curve under the influence of gravity? If the speed is
determined by gravity without friction, you can use conservation of energy to compute the speed. I’ll use the
coordinateymeasured downward from the top point of the curve, then

y

x

mv^2 / 2 −mgy=E, so v=

√

(2E/m) + 2gy (4)

Suppose that this particle starts at rest fromy= 0, thenE= 0andv=

√

2 gy. Does the total time to reach a
specific point depend on which path you take to get there? Very much so.
1 Take the straight-line path from(0,0)to(x 0 ,y 0 ). The path isx=y.x 0 /y 0.

ds=

√

dx^2 +dy^2 =dy

√

1 +x^20 /y^20 , so

T=

∫

ds

v

=

∫y 0

0

dy

√

1 +x^20 /y 02

√

2 gy

=

√

1 +x^20 /y 02

1

√

2 g

1

2

√

y 0 =

1

2

√

x^20 +y 02

√

2 gy 0

(5)

y

2 There are an infinite number of possible paths, and another choice of path can give x
a smaller or a larger time. Take another path for which it’s easy to compute the total
time. Drop straight down in order to pick up speed, then turn a sharp corner and coast
horizontally. Compute the time along this path and it is the sum of two pieces.

∫y 0

0

dy

√

2 gy

+

∫x 0

0

dx

√

2 gy 0

=

1

√

2 g

[

1

2

√

y 0 +

x 0

√

y 0

]

=

1

√

2 gy 0

[

x 0 +y 0 / 2

]

(6)