Answers to the Exercises - 3D Math Primer for Graphics and Game Development - page 787

(d) Assuming the spring and environment had not changed, we would expect a 1 kg
mass to cause an extension of 2 cm. Since the actual change in length was 8 cm,
there are only two explanations. 2 Either the spring constant has been reduced or the
apparent force of gravity has increased (or both). Maybe the spring was worn out?
The increase in gravity could be caused by conducting the experiment on a larger
planet or in a noninertial reference frame that is accelerating upwards.

√

1.00×10 2 (kg m/s 2 )/m
5.00 kg

1
2π

k
m =

1
2π

20.0 s −2
2π

7. (a) F = = = 0.712 Hz

(b) The amplitude is simply the initial displacement, 14.7 cm.
(c) We know that the motion of the mass must be of the form A cos(ωt + θ 0 ). We
already determined the amplitude A = 14.7 cm, and we know the angular frequency
ω = 2πF = 4.47 Hz.
When the mass crosses the rest position, cos(ωt + θ 0 ) = 0. Therefore, at this time,
sin(ωt + θ 0 ) = ±1, and the velocity is

v(t) = −Aω sin(ωt + θ 0 )
= ±(14.7 cm)(4.47 s −1 ) = ±65.7 cm/s

Since speed is always positive, we can discard the “±”.
8. Since there are no external forces, the center of mass of the man + car system does not
move, and the total momentum of this system must remain zero throughout. We'll let v m
and v c refer to the velocity of the man and car, respectively, relative to Earth.
(a) The relative velocity of the man and car is expressed by

v m −v c = 1.25 m/s,
v m = v c + 1.25 m/s,

and we also know that the combined momentum of the system must remain at zero,

P m + P c = m m v m + m c v c = 0.

Plugging the first equation into the second, we have

(75.0 kg)(v c + 1.25 m/s) + (1.00×10 3 kg)v c = 0,
(75.0 kg)v c + 93.8 kg m/s + (1.00×10 3 kg)v c = 0,
(1.08×10 3 kg)v c = −93.8 kg m/s,
v c = −0.0869 m/s.

So we obtain the inertial velocity of the man as

v m = v c + 1.25 m/s = −0.0869 m/s + 1.25 m/s = 1.16 m/s.

2 No credit is given for suggesting “physics stopped working.” However, if you answered “we are inside
of a video game,” give yourself 20 points extra credit.