The equation as to how far is the transmission necessary in a vehicle may be answered by considering :-

(a) Variation of resistance to the vehicle motion at various speeds.

(b) Variation of tractive effort of the vehicle available at various speeds.

Total resistance to the vehicle motion :-

It consists of

(i) Resistance due to wind :- This is taken to be proportional to the square of the vehicle speed.

(ii) Resistance due to gradient :- This remains constant at all speeds. This is the component of the vehicle weight parallel to the plane of the road.
(iii) Miscellaneous :- Apart from the about two types, various other factors also contribute towards the vehicles resistance. These are type of the road, tyre friction, etc. This may also be taken approximately to remain constant with the speed.

The total resistance for a particular type of road, therefore maybe represent as shown in figure.

The total resistance for same type of road with different gradient may then be represented by curves shown in figure. The higher curve represents steeper gradients.

Tractive effort :-

The curves 1, 2 and 3 respectively in figure represents the tractive effort in first, second and top gears respectively.

Transmission necessity :-

By now we understand the variation of total resistance to the vehicle motion and the tractive effort of the vehicle with speed. It is obvious that whenever the tractive effort exceeds the total resistance, the vehicle will accelerate to a speed where tractive effort becomes equal to the total resistance.

For further clarification, consider figure. This is obtained by superimposing figure 2 on figure 3. Let the vehicle be in the top gear and suppose the vehicle is travelling on a gradient which gives total resistance curve I. Then from figure it is seen that OA is the stabilizing speed. If the speed at any instant is less, say, OB, the excess of tractive effort will accelerate it to speed OA. Similarly if the speed at any instant is OC, the excess of resistance will decelerate it to OA.

Now let the vehicle go on next gradient of curved II. In this case it is noticed that that stabilizing speed has decreased. Next consider further the curve III. At this gradient, we see that nowhere does the curve 3 cross curve III. Therefore the vehicle will not be able to go at this gradient in the top gear. However, if we pass on to second gear, we get a stabilizing speed OD. Similarly in the second gear also the vehicle will not be running on gradient IV for which we shall have to shift to first gear.

Again at start more acceleration is needed to gain speed quickly. This can best be done in first gear because in this gear the maximum tractive effort is available for acceleration. However, when the necessary speed has been obtained, we may shift into higher gears, because then the vehicle speed has to be simply maintained and no acceleration is required.