Force required to drive car

In a perfect vacuum, on a frictionless road, you could just turn off the engine and the car would keep moving, never slowing down. However, in the real world, there are several effects that exert a force on a moving car, slowing it down, such as:

• rolling drag between the tires and the road surface,
• fluid drag from the air that the car moves through, and
• various friction losses between moving parts in the car itself, which, unless compensated by engine power, cause the wheels to slow down and exert a torque on the roadbed slowing down the car.

To keep a car moving at a constant speed, the engine needs to exert enough force to balance all these forces.

The important thing to realize here is that, at high speeds, the main force slowing down the car is actually the fluid drag, which grows roughly in proportion to the square of the speed. Thus, to double the speed, the engine needs to exert four times as much force. (At least, that approximately holds at normal highway speeds. Things get even more interesting when you approach the speed of sound and wave drag starts to play a role.)

Because of this, minimizing the drag coefficient is a critical feature of high-speed automobile design, and is why essentially all modern cars (but especially high-speed models) feature streamlined shapes designed to minimize aerodynamic drag.

Also, as Sachin Shekhar notes in his answer, the rolling drag for pneumatic wheels is also somewhat speed dependent, mainly because the wheels are flexible, and thus deform as they rotate, losing energy as heat. These losses also increase with velocity, meaning that, even in vacuum, maintaining a higher velocity still needs more power. In principle, you could minimize these losses by making both the wheels and the road surface as hard and inflexible as possible — say, by making both out of steel, as is done for trains, which also minimize aerodynamic drag by their long and narrow shape. That's one reason why trains can travel at considerably higher speeds than would be practical for a car to maintain.

(Of course, to reduce drag even further, you could put wings on the car and have it fly high above the road, eliminating rolling drag entirely and reducing fluid drag significantly due to the lower air pressure in the upper atmosphere. Or, even better, go even higher up where the air is even thinner.)

You are wrong at assuming constant friction. Rolling Friction increases when you increase speed of the car (See the formulae at the bottom).

Also, aerodynamic drag increases with the square of speed (See the formula at the bottom).

So, at higher speed, the car engine needs to counter higher rolling friction and air drag to maintain that speed.

While the previous reason is adequate, I should add this: At higher speed the car needs to do more work per unit time. At 10Km/h, if tyres are at x RPM, the tyres need to revolve at, say, 10x RPM to maintain 100Km/h. So, engines are doing 10x more work per unit time at 100Km/h.

But, rolling friction and air drag (most dominant forces among all resistances here) are at the heart of the problem. In vacuum and frictionless environment, for example, you can turn OFF engine at 10x RPM and it'd remain at 10x RPM.

Here's the formula of Rolling Friction..

$$F_r = c_r N$$

Where $c_r$ is rolling friction coefficient (dimensionless) and $N$ is the Normal Force applied by road which is equal to weight of the car.

Now, the Rolling Friction Coefficient increases with the car speed resulting in the increase of the Rolling Friction.

For example, the rolling friction coefficients for pneumatic tyres on dry roads can be calculated with

$$c_r = 0.005 + \frac {1}{p} (0.01 + 0.0095(\frac{v}{100})^2)$$

where $p$ is tyre pressure and $v$ is speed

Here's the formula of Automobile Aerodynamic Drag..

$$F_a = A/2 × c_d × D × v^2$$

where $A$ is frontal area of the car, $c_d$ is drag coefficient, $D$ is density of air, and $v$ is speed.

Note: Weight of car goes down when gasoline burns which contributes in decrease of the rolling friction and inertia (Rocket Science), but resistance forces win by very large amount.