Cars have what is called a brake booster which use force from the engine to help you apply the brakes. Traditionally this used a vacuum servo connected to the engine air manifold but in cars without a carborator they need a dedicated vacuum pump powered by the engine. In more modern cars they may have implemented a similar system that does not use vacuum in order to avoid the dedicated vacuum pump. But they still have a brake booster as well as a steering servo. So if the engine is off these are not powered and the car becomes hard to control.
There is a brake assist drum near the engine. It helps with braking by using the vacuum from the intake manifold of a running engine. If you look on this drum, which is located under the brake fluid reservoir, you’ll see a tube connecting the drum to the intake manifold. If the engine isn’t running, it doesnt’t cause a vacuum, so there is no brake assist.
There’s a brake booster that sits in between the actual lever you press against when you hit the brakes and the hydraulic pistons that deliver braking power to the wheels. It uses vacuum from the engine intake system to basically preload a spring that helps you put more pressure on the brakes.
So if your engine isn’t running, it’s not creating an intake vacuum, which means the booster has nothing to work with.
That’s a very simplified version. This is slightly better: https://www.youtube.com/watch?v=wbTUvp-tD5M
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