Answering “why” this happens is actually pretty straight forward.
A spinning/sliding tire has significantly less grip than one that isn’t spinning/sliding. When doing something like a burnout, the tires are spinning, there’s much less grip influencing the direction of the car.
The spinning tires always want to be in front. No problem with a FWD car as it basically just self-centers, but in a RWD car, it means the car wants to spin around. A skilled driver can keep that spin from happening by steering into it in a controlled slide.
Mustangs are famous for doing burnouts with ease, but then the automatic transmission decides it’s time to shift. That shift causes a loss of power, a loss of spinning, and the grip suddenly being regained. The controlled slide is lost, the car shoots off in whatever direction it’s pointing.
Then, when the next gear engages, the tires break loose again. The car now has inertia from grip, but no grip to counteract it, so it’s extremely difficult to keep under control. This is why you can so often hear the gear shift occur in those videos right before the loss of control.
Preventing this is easy, just don’t do burnouts in automatic mode.
If you accelerate, the acceleration force must be transfered by the wheels to the road.
This is done through the friction of the tire with the road.
If you try to accelerate more than you have friction available, the tires start to spin and you lose all friction on the spinning wheel(s).
Loss of friction means the wheel is also free to slide sideways.
As the car is a rear-wheel drive, the loss of friction will be on the rear wheels, meaning the backside of the car is free to slide sideways.
Because there is still friction on the front wheels, any slight inbalance in force during this acceleration will make the rear of the car go sideways in one direction.
This motion has a possitive feedback loop, so it gets worse quickly up to a point the car can no longer be controlled by the driver.
For rear-wheel driven cars, preventing it is done by keeping accelerating under control by the driver, or by electronic or mechanical aids that keep the spinning limited per wheel.
As an opposite case: the reason Teslas dominate at starting in a strait line (even the RWD only model) is because electric motors are so good at getting that power to the ground (i.e. modulating / controlling the power going to the wheels) that wheel slippage doesn’t occur (recall static friction > sliding friction), where as a gasoline engine is more brute force and so any traction control would need to be done by the brakes.
First: they are bad drivers with more money than sense.
But in regards to muscle cars, they have heavy, powerful engines at the front and they deliver their power to the rear wheels. That means most of the weight is on the front tires, and the rear tires have less adhesion to the road. As a result, if you simply go full throttle from a standstill, the energy the engine sends to the rear tires far exceeds the grip they are capable of and the wheels start spinning out. Since the two tires are constantly experiencing changing and differing levels of grip, the back of the car tends to fishtail as one tire experiences slightly more grip than the other.
The solution to this depends on what your initial goal is. If you’re in a drag race, the solution is to ease of the throttle and find the point where the tires are able to grip the road and accelerate properly. If you’re doing burnouts or drifting (IE making the wheels spin on purpose) the solution is to subtly steer towards the side of the car that is sliding away countering a slide. Drifting essentially involves doing this through the entire turn.
Both of these are basically subconscious actions from a professional driver and they can do it almost without thinking. But the videos you see on YouTube of car meetups and street takeovers are usually novice drivers who have a lot of disposable income to invest in a car and attempt burnouts and donuts that seem easy but actually require quite a bit of skill.
A small point to add:
The modern Mustang (or any modern car, really) will not let you get too rowdy unless you turn traction control off. All those idiots swaying into traffic have made a deliberate choice to turn off “the electronic nanny” and put their lack of skill on display for everyone. You can squeal the tires with traction control on, but once you start swaying it cuts power fast.
Solid axle is a little jumpier than independent, but it’s only a little. When you’ve lost control an independent suspension (like the Mustang has had since 2014) isn’t going to save you.
An open diff (as opposed to any form of posi/locker) is generally not going to let you do this as much and will just spin one tire in place. Most performance cars will have some form of a limited slip or locking diff so that they can put power down with two wheels. When both wheels start to slip they will slip unevenly (as others have covered well in this thread.)
These cars are powered by the back wheels. When you punch the gas, two things happen.
1. The back end of the car will kick to the side a little bit before traction is gained which is completely normal.
2. The sudden burst of speed causes the car body and frame to tip back a little bit which puts more down force on the back tires and gives more traction. This lets the tires bite into the ground and stop kicking out to the side.
An experienced driver will maintain acceleration along with continuing to look and steer in the direction they want to go. The back end will quickly get back in line and off they go.
An inexperienced driver will feel the back end kick to the side and panic. They will over steer and let off the gas. This allows three bad things to happen.
1. The car is now decelerating which makes the frame and body tip towards the front. This removes the down force from the back tires and applies it all to the front tires.
2. The back tires don’t have traction and are now unable to do anything. Even if the driver hits the gas again, they will just spin and kick the back end out more.
3. All of the down force and traction are running through the front tires which have been over steered. We now have very grippy tires steering the car too far to the side.
The car is now in an uncontrollable fish tail that will result in a crash in anything other than an empty parking lot.
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