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Troublshooting your pc for dummies - Bourg D.M.

Bourg D.M. Troublshooting your pc for dummies - Wiley publishing , 2002. - 350 p.
ISBN 0-596-00006-5
Download (direct link): fordummiestroublesho2005.pdf
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Consider the collision between the club head of a golf club and a golf ball as illustrated in Figure 5-6.
In the velocity diagram on the left, v_ represents the relative velocity between the ball and club head at the instant of impact, v+ represents the velocity of the ball just after
98 | Chapter 5: Collisions
n
Figure 5-6. Golf Club-Ball Collision
impact, and vt_ and vc+ represent the tangential components of the ball velocity at and just after the instant of impact, respectively.
If this were a frictionless collision, vt_ and vt+ would be equal, as would the angles a and 9. However, with friction the tangential velocity of the ball is reduced, making vt+ less than vt_, which also means that a will be less than 9.
The force diagram on the right in Figure 5-6 illustrates the forces involved in this collision with friction. Since the ratio of the tangential friction force to the normal collision force is equal to the coefficient of friction, you can develop an equation relating the angle <p to the coefficient of friction:
In addition to this friction force changing the linear velocity of the ball in the tangential direction, it will also change the angular velocity of the ball. Since the friction force is acting on the ballís surface some distance from its center of gravity, it creates a moment (torque) about the ballís center of gravity that causes the ball to spin. If you use an approach similar to the rolling cylinder example back in Chapter 4, you can develop an equation for the new angular velocity of the ball in terms of the normal impact force or impulse:
tan <f> - Ff/Fn = fi
Notice here that the integral on the left is the normal impulse; thus,
Impulse ó Icg/(fir){co+ ó co-)
CO+ = (Impulse) (jLtr)/ ICg + CO-
A,
Friction 99
This relation looks very similar to the angular impulse equation that I showed you earlier in this chapter, and you can use it to approximate the friction-induced spin in specific collision problems. "T
Turn back to the equation for impulse, J, in the preceding section that includes both linear and angular effects. Here it is again for convenience: x ? s o'y,
J = óvr(e + l)/[l/mi + l/m2 + n ē (ri X n)/Ii + n ē (r2 x n)/I2]
This formula gives you the collision impulse in the normal direction. To see how friction fits in, you must keep in mind that friction acts tangentially to the contacting surfaces, that combining the friction force with the normal impact force yields a new effective line of action for the collision, and that the friction force (and impulse) is a function of the normal force (impulse) and coefficient of friction. Considering all these factors, the new equations to calculate the change in linear and angular velocities of two colliding objects are as follows:
vi+ = vi_ + [Jn + [fi] )t]/mi
v2+ = v2_ + [-Jn + (pj)t]/m2
ui+ = + {n x [Jn + faj)t]}/Icg
tu2+ = u2~ + {r2 x (-Jn + (fij)t]}/lcg
In these equations, t is the unit tangent vector, which is tangent to the collision surfaces and at a right angle to the unit normal vector. You can calculate the tangent vector if you know the unit normal vector and the relative velocity vector in the same plane as the normal vector:
t = (n x vr) x n l^yke. Vi * ( h * 'Jr ) t = t/|t|
For many problems that youíll face, you may be able to reasonably neglect friction in your collision response routines, since its effect maybe small in comparison to the effect of the normal impulse itself. However, for some types of problems, friction is crucial. For example, the flight trajectory of a golf ball depends greatly on the spin imparted to it as a result of the club-ball collision. Iíll discuss how spin affects trajectory in the next chapter, which covers projectile motion.
100 | Chapters: Collisions
CHAPTER 6
Projectiles
This chapter is the first in a series of chapters that discuss specific real-world phenomena and systems, such as projectile motion and airplanes, with the idea of giving you a solid understanding of their real-life behavior. This understanding will help you to model these or similar systems accurately in your games. Instead of relying on purely idealized formulas, Iíll present a wide variety of practical formulas and data that you can use. Iíve chosen the examples in this and the next several chapters to illustrate common forces and phenomena that exists in many systems, not just the ones Iíll be discussing here. For example, while Chapter 8 on ships discusses buoyancy in detail, buoyancy is not limited to ships; any object immersed in a fluid experiences buoyant forces. The same applies for the topics discussed in this chapter and Chapters 7, 9, and 10.
Once you understand whatís supposed to happen with these and similar systems, youíll be in a better position to interpret your simulation results to determine whether they make sense, that is, whether they are realistic enough. Youíll also be better educated on what factors are most important for a given system such that you can make appropriate simplifying assumptions to help ease your effort. Basically, when designing and optimizing your code, youíll know where to cut things out without sacrificing realism. This gets into the subject of parameter tuning.
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