Optimizing Car Performance: Power, torque, rpm

[FrvrUndd]

You want to select a gear such that the rpm is low enough to avoid overheating and you want to choose a gear that gives you enough torque for the application.
A car is going up a 20% grade at a speed of 60 km/h.

Gear ratios:
1st - 3.143:1
2nd - 1.870:1
3rd - 1.235:1
4th - .949:1
5th - .727:1
Reverse - 3.308:1
Differential Ratio (final drive ratio between axle and transmission gear) - 4.2941:1
Wheel diameter = 16in
Mass=1218 kg
Tachometer redline = 6800 rpm


Questions:

If the coefficient of static friction btween the tires and the ground is .9, what is the minimum power required to keep the car moving at the same speed?

What is the minimum torque required on the wheel?

If the driver chooses to go up the grade in fifth gear, what is the rpm of the engine? What is it for first gear?

The torque output by the engine is related to the force produced by the piston upon ignition of the gas and varies with the angular velocity of the engine. Assume that the engine torque is giving by
t(engine)=Aw^2(engine)+Bw(engine)+C, where A=-.0004 Nms^2/rad, B=.731 Nms/rad, and C=16.789. Given this information, what is the torque on the wheels in fifth gear? What is it in first gear?

Given the torque requirements and the fact that you do not want to overheat the engine, what is the gear you should use?

 

[jbsteele]

Answer:The minimum power required to keep the car moving at the same speed is 24.3 kW. The minimum torque required on the wheel is 893.5 Nm. If the driver chooses to go up the grade in fifth gear, the engine rpm is 2265 rpm. For first gear, it is 6778 rpm.The torque on the wheels in fifth gear is 705.8 Nm and in first gear is 2099.7 Nm. Given the torque requirements and the fact that you do not want to overheat the engine, the best gear to use would be fourth gear as it has a lower rpm than fifth gear and gives enough torque for the application.