Torque, Horsepower, Acceleration

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1	This spreadsheet shows why torque and gearing are the important factors in vehicle performance. Many people think horsepower
2	is the important factor.
3	All the formulas are visible in the cells, they are all straight high school physics. As is generally the case in examples like this, we
4	are assuming zero mechanical and aerodynamic drag. (You can plug different numbers in for stuff, but be careful not to mess up formulas. If you do you can just make a new copy).
5
6	We have a motorcycle with the gear ratios, weight and drive wheel radius listed. We know that when accelerating at WOT the
7	engine produces 50 foot-pounds of torque at 5,000 RPM. Let's see what's happening at the rear wheel in different gears.
8
9	First, how much horsepower does 50 ft-lbs of torque at 5,000 RPM represent?
10	HP = torque * RPM / 5,252, and 50 * 5,000 / 5,252 = 47.6 HP. That's the short formula. There's some stuff simplified in there.
11	The "real" formula looks like this:
12	HP (ft-lbf/min) = {torque (ft-lbf) * (rev/min) * (2 * Pi) (rad/rev)} / [33,000 {(ft-lbf/min)/(HP)}]
13
14	Let's look at acceleration potential in each gear at 5,000 RPM, WOT.
15								Rear wheel torque "on the ground"
16								Foot pounds force
17	Gear ratios		Gear	Overall	RPM	Torque	Horsepower	1	2	3	4	5	6
18	Primary	2.09	Ratio	Ratio	5,000	50.0	47.6	771.0	556.9	432.6	378.5	301.2	266.8
19	First	2.46	15.42	15.42
20	Second	1.78	11.14	11.14				Acceleration, foot/second^2
21	Third	1.38	8.65	8.65				38.2	27.6	21.4	18.7	14.9	13.2
22	Fourth	1.21	7.57	7.57
23	Fifth	0.96	6.02	6.02
24	Sixth	0.85	5.34	5.34
25	Final	3.00		Overall ratio includes drive wheel radius
26
27	Vehicle weight =		650.0	pounds	Includes rider
28	Drive wheel radius =		12.0	inches
29
30	What is the HP "on the ground" in each of the gears at this time? Using HP = torque * RPM / 5,252 we can calculate HP.
31	The RPM of the rear wheel is (engine RPM / (gear ratio)* (12 / drive wheel radius) (corrects torque for the wheel radius).
32
33	Rear wheel HP in gear
34	1	2	3	4	5	6
35	47.6	47.6	47.6	47.6	47.6	47.6
36
37	The HP is the same in all six cases, identically engine horsepower. So calculating acceleration from HP is not possible without first
38	converting HP to force (in this case, torque).
39
40	Here is a torque/HP graph that further illustrates the lack of relationship between HP and acceleration. It's the same vehicle above
41	with an unusual engine that makes a constant torque. Let's look at first gear acceleration.
42
43	RPM	Torque	HP	Acceleration
44	1000	50	9.5	38.2
45	2000	50	19.0	38.2
46	3000	50	28.6	38.2
47	4000	50	38.1	38.2
48	5000	50	47.6	38.2
49	6000	50	57.1	38.2
50	7000	50	66.6	38.2
51	8000	50	76.2	38.2
52	9000	50	85.7	38.2
53	10000	50	95.2	38.2
54
55
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62
63	HP continuously increases, torque remains constant, acceleration remains constant.
64
65
66	Here's another example with a "special" engine. In this case HP remains constant.
67
68	RPM	Torque	HP	Acceleration
69	1000	131.3	25.0	100.3
70	2000	65.7	25.0	50.2
71	3000	43.8	25.0	33.4
72	4000	32.8	25.0	25.1
73	5000	26.3	25.0	20.1
74	6000	21.9	25.0	16.7
75	7000	18.8	25.0	14.3
76	8000	16.4	25.0	12.5
77	9000	14.6	25.0	11.1
78	10000	13.1	25.0	10.0
79
80
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86
87
88	Torque and acceleration are again proportional, HP again is not related to acceleration.
89
90	In this example torque (and so acceleration) decreases. Horsepower increases.
91
92	RPM	Torque	HP	Acceleration
93	1000	50.0	9.5	38.2
94	2000	48.0	18.3	36.7
95	3000	46.0	26.3	35.1
96	4000	44.0	33.5	33.6
97	5000	42.0	40.0	32.1
98	6000	40.0	45.7	30.6
99	7000	38.0	50.6	29.0
100	8000	36.0	54.8	27.5