The car and everything in it is all travelling at the same velocity when it is in normal motion, however when the brakes are applied they are only working to stop the velocity of the car, the brakes don’t stop the velocity of the passengers so you “jerk forward” until the seatbelt or something else stops you.
The inertia of you is the same as the inertia of the car. So when you decelerate your body still wants to go the speed you were going before braking. So if the car was going 80mph(100kph) and you slowed down, your body will have a negative g response pressing into the seatbelt as your forward inertia dissipates.
The brake pads clench on the tires and stop them, but a vehicle has shocks that allow the vehicle to lurch forward slightly and then correct itself backwards, instead of continuing to put stress on the break pads and wheel wells. This allows for faster stops when necessary, and lessens wear and tear on various vehicle parts.
PS if you lurch hard every time you stop (and don’t have a REALLY old vehicle), you need to work on your situational awareness and response time or get your vehicle looked at. That’s not normal to lurch hard every time you brake.
Velocity is relative, acceleration isn’t.
So, looking at this situation from the perspective of someone on the ground, you and the car are both moving forward at the same velocity, until the car suddenly starts braking. You, having momentum, continue moving forward until friction or another force changes your velocity to match the car’s new velocity.
Looking at the situation from your perspective inside the car, you and the car are both stationary, the rest of the world moving around you. When the car starts braking, it starts moving backwards. You, having momentum, continue to be stationary until friction or another force changes your velocity to match the car’s new velocity.
Actually it is moving from your perspective, the act of you jerking forward is movement relative between you and the car. This is caused by the cars acceleration (or deceleration in this case) a force is applied to the car(breaks) and is slowing the car down but since you are not structurally connected to the car there is no force being applied to you until the seatbelt, as well as friction between you and the seat, applies a force on you slowing you down.
The car’s various parts also experience the forces from acceleration and deceleration. Just that it’s all bolted together so it’s much less likely to jostle around like you would, but you will notice that a car will look to be leaning forward on a hard stop or back on hard acceleration in relation to its wheels (which are also experiencing forces).
You and the vehicle are most definitely moving from the perspective of the Earth and gravity.
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