thehog said:
All very well but perhaps explain why you think this is so, rather than "I'm a student of physics".
Your input would be most welcome.
GoodTimes said:
Fair enough. I thought about my post, and find it perhaps hypocritical as I've always hated arguments from authority....
Let me define how I understand the positive argument that Ryder's bike was powered is constructed.
Let RB represent Ryder's Bike.
1. Ryder Crashed.
2. The behavior of RB during / after the crash was unusual:
a) RB came to a stop.
b) Ryder reached for RB.
c) Ryder was unable to grab RB, as RB accelerated.
d) Objects at rest tend to stay at rest...
e) RB did not stay at rest. That is unusual.
3. The best (or only?) explanation for this behavior is that the rear wheel was powered during the crash.
4. Since the wheel received power, and Ryder was not pedaling, the the only possible source of power to the rear wheel is a motor.
I see problems with many of the above points.
To 1: I do not dispute
I take exception to 2, 3, and 4.
Before I invest too much time, I'll let the above simmer. If somebody wants to modify, or correct my understanding, please feel free.
to 3: To my knowledge, there is no documented case of a pro being caught with a motor in their bike. The burden of proof is on the person who wants to defend the motor hypothesis. As can be shown below, there are rational explanations that do not require a motor.
to 2:
In summary, I believe that it's possible that Ryder's back wheel maintained a portion of it's angular momentum during the crash. Then, a small amount of friciton between wheel and pavement caused the bike to accelerate away from ryder.
1. Let t_0 represent the moment when Ryder loses control and begins to crash. Assume Ryder is moving at 60 kph (v_0) at time of crash [conservative, they are probably going faster].
2. Assume wheel diameter of 0.7 m (D_bw), radius of 0.35 m (r_bw), wheel mass of 0.9 kg (m_bw), and moment of inertia of 0.1 kg*m^2 (I_bw) [conservative estimate. I do not feel like experimenting to determine actual value. you can see a discussion here:
http://www.physicsforums.com/showthread.php?t=586625]
3. Assume bike has mass of 10 kg (m_b), center of gravity of bicycle is in geometric center of bicycle. This is not very important, but comes in a bit later on.
4. From 1 and 2: angular velocity of wheel at time of crash:
60 kph = 16.7 m/s
w_bw_0 = v_0/r_w
16.7 m/s / 0.35 m = 50 rad/s
5. Energy of back wheel at time of crash (E_bw_0): From 4, and 2:
E_bw_0 = 1/2 * I_bw * w_bw_0^2
E_bw_0 = 1/2 * (0.1 kgm^2)(50 rad/s^2)
E_bw_0 = 125 J
The above should not be contentious. This is just the kinetic (rotational) energy of the back wheel of Ryder's bike when he lost control and crashed, and some reasonable assumptions regarding bike geometry and mass. # 8, below, is the most liberal assumption that I'm going to make in this analysis. I think the assumption is plausible, which is all I think I need to discount the motor hypothesis.
6. Let the moment when ryder unclips his foot be referred to as t_1.
7. From the video, it appears that Ryder and his bicycle come to almost a complete rest at t_1. Hence, let v_1 = 0 km/h.
8. From watching video, it seems plausible that the back wheel of Ryder's bike did not make strong contact with pavement before he unclipped his foot.
8a. I note that for some time, his wheel did drag on the pavement sideways, so let's assume that the back wheel of ryder's bike lost half of it's angular velocity between t_0 and t_1.
8b. From 8a,
E_bw_1 = 30J.
w_bw_1 = 25 rad/s
also, let P_1 be momentum of system at time t_1. P_1 = I_bw*w_bw_1, = 2.5 Js.
9. Assume COF of wheel to road is 0.5.
10. (to derive the normal force of the back wheel on the road, we divide by 2, since we assumed COG of bike in geometric center. therefore back wheel is supporting 1/2 the mass of the bike)
Normal force of back wheel on road:
Fn_bw_1 = m_b *g / 2
= 10 kg * 10 m/s2 / 2
= 50N
11. force of friction of back wheel on road:
Ff_1 = FN_bw_1 * COF
= 25N
12. Linear acceleration of bicycle, caused by energy contributed by back wheel:
a_b_1 = Ff_1 / m_b
= 2.5 m/s2
Hence ryder's bike accelerated away from him. From the video evidence, it seems that my proposed rate of acceleration is within the right ballpark.
Let t_2 be the time when moto ran over ryders bike.
Equations that model the complete acceleration of Ryder's Bike between t_1 and t_2 go beyond the scope of this analysis. It is complicated by the fact that the back wheel has a discrete amount of energy, and so the force that it affects on the bicycle reduces with time. There are a few sanity checks that we can do though, to help show that the above analysis is valid.
I need to get going, but one easy sanity check I'll offer is a look at conservation of momentum.
Assume between t_1 and t_2, 25% of the systems momentum is lost to friction between bicycle and road.
Therefore, P_2 = .75* P_1 = 2 Js.
note: between t_1, and t_2, bicycle is pivoting on handlebars. Therefore only half of bike is moved by wheel, hence m_b/2:
V2 = P2 / (m_b /2)
= 2 kg m/s / 5 kg
= .4 m/s
= ~ 1.5 km/h
When ryder's bike gets hit, it's not going very fast. Is the rear wheel going faster than 1.5 km/h? maybe a bit. But a lot of my assumptions (especially, regarding loss of energy between t_0 and t_1) have been conservative.
It should be noted that I graduated a few years ago, and real - world engineering involves frighteningly little actual calculation and math, so I'm a bit rusty! I have not done a thorough review of the above, so if theres a typo or a small calculation mistake, please don't crucify me
