**sixth power**of the EV speed (v), so (constant K1 mainly depends on particular EV model):

Pb = K1*v^6

This means that if the EV travels at 60 mph (96 km/h) and the dissipated power in the battery is for example 1 kW, the dissipated power at 70 mph (112 km/h) will be 2.5 kW and the dissipated power at 50 mph (80 km/h) will be 335 W.

By the way, the cumulative dissipated energy in the battery on the fixed travelled distance (Eb with units in e.g. kWh) is proportional to the fifth power of v:

Eb = K2*v^5

If the temperature of protective case for the battery is more or less equal to the environment temperature (Te), the final temperature difference (after long long time!) between the battery (Tb) and the environment should be proportional to the battery dissipated power:

Tb – Te = K3*v^6

So, for example, if the battery temperature at

**60 mph**is

**6oC**higher than the environment, at

**70 mph**it will be about

**15oC**and at

**50 mph**about

**2oC**higher.

Note that short-term temperature changes at some fixed travelled distance depend on the fifth and not the sixth power of the speed. Therefore, if the battery is at environment temperature and we travel some shorter distance at 60 mph and the battery temperature rises for 6oC, at 70 mph, on the same distance, it will rise for about 13oC and at 50 mph for about 2.4oC.

Therefore, even slight changes in EV velocity can result in larger changes of dissipated power inside the battery (and hence in larger temperature changes).

**Don’t drive too fast!!**

By the way, the above dissipated power expression is derived from the following relations:

- At higher speeds the EV power is proportional to the third power of velocity (by the way, the average consumption (in kWh / mile or km) is proportional to the square of velocity). The current through the battery can be assumed to be proportional to the power delivered to the EV motor (due to more or less constant voltage of the battery), so the current through the battery is proportional to the third power of the EV velocity.

- Due to internal resistance of the battery, the dissipated power in the battery is proportional to the square of the current. Therefore, the dissipated power in the battery is proportional to the sixth power of the EV velocity.

Note that the above derivations are approximations and I did not want to make the main idea too complicated.