How Tanks Move

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The How
  Tanks use high power engines to turn a toothed metal wheel called a sprocket. These sprocket teeth connect to corresponding grooves in the tanks' linked metal tracks which the tanks road wheels run along. This system of tracks and wheels offers more traction than standard wheels. Instead of the wheels having traction on the ground in 4 relatively small places, a tank has traction along the entirety of its track, meaning that there is more surface area for the tank to grip, lessening the likelihood of getting stuck, sinking, or losing traction. Modern tanks even have slots in the track to attach things like rubber pads so they can drive on roads, or teeth so they can better grip ice or loose ground. 

--In the picture below, you can see the metal sprocket to the rear of the tank. You can also see the grooves in the track and the road wheels that run along the track.

tank treads

Image source: I took this picture at the sword beach museum in Normandy

    The Physics
      When designing a tank, specifically the track system and the engine, engineers have to worry about how heavy the vehicle is so that they know how wide to make the tracks, what size they can make the wheels, and how many they need to have to adequately spread the load. 
FBD     
I've drawn a free body diagram (above) to help describe the physics. Here you can see that the sprocket exerts a moment on the track turning it and creating a force in the positive X direction. Then, depending on how much power the engine is supplying, the tank will overcome the force of friction and move forward. In the Y direction, the tanks' tracks need to be wide enough to displace the force of gravity so it does not sink on soft ground and can get enough traction.