I understand infantry is mechanized and only travels on foot when in actual combat. However tanks travel faster than people can walk and run. How do tanks and infantry support each other without tanks being vulnerable by moving slow, given that tanks are also vulnerable without infantry support?
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Depends on your usage of the tank. How does a person keep up with a vehicle in a parade? Same idea.
It mainly depends on the type of combat. Ie urban, desert, etc. But they can support each other from afar. The tanks big weakness is its lack of visibility (both ground and air) of its surroundings when in full combat mode, so infantry can act as spotters and target painters and take out air threats. Tanks can act as the big guns and take out approaching threats to infantry. In urban combat, tanks and infantry do stay closer together and the tanks provide cover.
If it’s in the open fields then tanks rush around and find hull down positions and fire at targets. Infantry will be travelling in vehicles to keep up.
If it’s in an area where tanks might be ambushed by infantry with hand held anti tank weapons, like in a city, then the infantry move ahead of the tanks and the tanks follow slowly behind.
In theory. US tanks seem to rush ahead in towns and cities without infantry support in a lot of combat vids but then again basically almost nothing can actually destroy them.
The Russians have been doing it in vids in Ukraine but they seem to be able to be blown up by everything.
There are of course many ways of using tanks and infantry together. Tanks do not usually move at full speed during the entire battle. It is generally better to move fast into cover and then use the stationary possition to better scout and aim at targets. Once there are no more targets you move forwards to the next cover. At the same time the infantry are doing the same but are able to move more constantly then a tank. The pace of both tanks and infantry is based on how quickly they can push the enemy back. Of course you can get into issues where tanks may overextend beyond the infantry but this leaves them vulnerable to enemy infantry traps so it is best for the tanks to hold back and wait for the infantry.
You can move the tank fast if it is supported by dismounted infantry.
One option is to design tanks that have a low max speed and increase the armor of the tank. That was done between the world wars and you end up with tanks like [https://en.wikipedia.org/wiki/Matilda_II](https://en.wikipedia.org/wiki/Matilda_II) with a top speed on-road of 24km/h and 14km/h off-road. It had armor that with a thickness up to 78mm and other tanks in the early part of the war could not take it out. During the https://en.wikipedia.org/wiki/Battle_of_Arras_(1940) Germany had to use the 88mm anti-air/anti-tank guns to take the mount.
The British also had fast but lighter armored cruiser tanks for exploration behind enemy lines.
Later in WWII and after it it was clear that that was not the best idea and you could build well armored and fast Main Battle Tank, this is what we have today.
Today infantry is transported in IFV and APC so they can move as fast last the tank if they are in them. If they are dismounted you need to have a slower avrage speed.
A tank that is still or moves slowly is not necessary a vulnerable tank. A tank that moves fast is not necessarily a take that is safe. If it is still or moving on an open field with enemies that are in range and are not detected is vulnerable. If a tank is stationary or drives in a location where the enemy can see it is safe.
So you can have that is in a turned town position behind the crest of ground. That means only sensors on the turret are exposed. You can then move forward engage the enemy quickly and move back in cover. You can be completely behind the crest and only move forward you get information about a target from someone else like infantry on the ground https://en.wikipedia.org/wiki/Hull-down
You can see a tank during something like this in https://www.youtube.com/watch?v=m9euAYAUyGs It is exposed for quite a long time but still, it manages to fire and back away before an anti-tank missile fired at it reach the tank.
So stay in an overwatch position like that and have infantry advance in front of you. Engage target as required.
Tanks should not operate alone so when some tank is in a position like that ready to engage the enemy other tanks can quickly drive forward to a new position where have some protection and can engage the enemy.
Advancing with your friend covering you is what infantry too.
The tanks stay with the people, not the other way around. For other situations the infantry follows with vehicles.
Tanks are usually paired with mechanized (tracked APCs) or motorized (trucks) infantry. During a movement-to-contact, the force will normally be organized into a recon element, advance guard, main body, rear guard, and screening elements. Most of the tanks and infantry are in the main body, with maybe a platoon size element of each in the advance and rear guards. If an element of the force makes contact with the enemy, then the whole force will orient on the enemy to give battle. This is when the infantry will dismount to maneuver on the enemy force with tank support.
If they are assaulting a pre-planned objective, then tanks will usually be a part of the support-by-fire position that suppresses the objective to allow the infantry to maneuver close enough to the objective and dismount in order to assault on foot.
Tanks generally don’t move with the infantry in combat. Mechanized infantry will be at the front supported by their IFVs while the MBTs provide fire support from a distance.
vehicles like the Bradley (for a great look at why the Bradley is like it is, the 1998 HBO movie [The Pentagon Wars](https://www.imdb.com/title/tt0144550/?ref_=fn_al_tt_1) (based on the real account of testing the prototypes). Also, the reason the Humvee is the size and design it was, is because it was designed to have the same track (wheel width) as the tanks, so it could drive in their tracks (and avoid anti vehicle mines).
They’re not left to run around/behind the tanks like [north korean bodyguards](https://www.youtube.com/watch?v=u8y8ndgk9DU&ab_channel=TheTelegraph) with Kim Jong Un’s limo.
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