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V

P

1

P

2

V

Q

P

2

Q

1

Figure 12.7.
Detecting a collision between two moving spheres.

()

()

PPV

Q

t

=+

=+

t

1

P

t

QV
.

t

(12.29)

1

Q

Let
r
and
r
be the radii of the two spheres. We wish to determine whether

the distance
d
between the centers

()

()

P

t

and

t

is ever equal to
P

rr

+

at some

Q

Q

)

time

. If so, then the spheres are tangent to each other at time
t
, and a col-

lision has taken place. We examine the squared distance between

∈

0,1

t

[

()

()

P

t

and

t

Q

given by

2

PQ
.

()

()

2

(12.30)

d

=

t

−

t

()

()

Substituting the values given by Equation (12.29) for

P

t

and

t

, we have

Q

2

d

=+ −−

PVQ V
.

t

t

2

(12.31)

1

P

1

Q

For convenience, we define

AP Q

BV V

=−

=−

1

1

(12.32)

P

Q

so that Equation (12.31) can be written as

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