When I was at school trying to work out at what rate water had to flow through a hole in the ceiling if the bath upstairs, containing 25 gallons, took 10 minutes to empty, my main concern wasn’t about the rate; it was why there was a hole in the ceiling.

My maths and physics teachers would despair.

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I eventually learnt that power = force x velocity… or something like that.

Thankfully, things have moved on in my insignificant world. I now know that power = a stonking great engine, turbos, superchargers and as much pressure as you can possible force into a variety of cylinders.

Audi has its RS models, BMW has its M cars and Jaguar has its SVR. Meanwhile, over at Mercedes-Benz, the equation is a simple one. Power = AMG.

First of all, beware the office braggart who announces to anyone who’s willing to listen that they have now acquired an AMG Mercedes. Yes, most of the Mercedes range is available with a high performance AMG version, but you can also get a humdrum model with an ‘AMG Line’ bodykit. For example, fork out £20,715.00 for an entry-level Merc A160, add a further £2,290, and you end up with an AMG body kit and 18-inch alloys. Inside you’ll be informed that you also have sports seats and brushed steel pedals.

However, you don’t get the interesting oily bits.

That’s why we love the Mercedes AMG A 45 4MATIC, although you’ll now need to find £41,875 before you dip into the options catalogue or £44,780 if you add the Aerodynamics Package which gives you the rear aerofoil and a large front splitter. It looks great in a aggressive, punch-you-in-the face sort of way.

The aforwmentioned power comes from a 376bhp four-cylinder 2.0-litre turbocharged petrol engine, which currently rates as the most powerful production car engine in the world in terms of the bhp:litre ratio. All of this means you’ll hit 60mph in a shade over four seconds. Top speed is restricted to 155mph. The brakes are terrific and quickly shave off speed. Go easily and you’ll manage just over 40mpg. Unlikely, we know, but we thought it was worth mentioning anyway.

All of this power goes through an AMG Speedshift 7-speed twin clutch sport auto transmission and is planted on the road via four-wheel drive. Although, under steady driving, the car defaults to front drive only.

As for the suspension set up, it’s up to you. There’s a knob situated behind the gearlever which allows you to use a Dynamic Select system offering four driving modes - Comfort, Sport, Sport+ and Race. In other words, choose the best ride and handling characteristics according to road (or track) conditions, or your mood.

Thanks to the excellent all-wheel drive system, this AMG A 45 can corner at astonishing speeds. You need to be doing something really stupid to make it come unstuck

Comfort mode delivers smoother gearshifts, a smooth ride, better economy and a low-key exhaust note from the twin pipes. Sport mode brings quicker engine and transmission responses, firmer suspension and a fruitier soundtrack. Moving on to Sport+ takes this a step further with quicker shift times and throttle responses, uprated ride quality and throttle blips on down-changes. Race mode is what it says on the tin.

OK, the Sport and Race modes are fine for a bit of fun but, after a while of being jiggled around on a long motorway journey, the enjoyment starts to wear a bit thin and most folk will happily revert to the Comfort setting, which is surprisingly forgiving. It soaks up speed bumps, ruts in the road, and actually makes this blisteringly quick A 45 a bit of a pussycat.

Move away from the mechanical stuff and you end up with a thoroughly useable five-door hatchback. The multi-adjustable, heated sports seats are brilliant providing that happy mix of comfort and support. There’s plenty of kit which comes as standard, too, including sat nav, air con, reversing camera, terrific audio system and stacks of neat AMG touches to dials and air vents. There’s no need to dip into the options list, really.

It also scores highly when it comes to safety equipment.

So, what’s it like to drive? The bottom line here is that, thanks to the excellent all-wheel drive system, this AMG A 45 can corner at astonishing speeds. You need to be doing something really stupid to make it come unstuck. In a straight line, it’s up there with cars like the Porsche 911, which is saying something. Choose your moment in a corner, floor the throttle and you’ll hurtle out of the other side. The levels of grip are stunning.

It might not have the burble of a BMW six-pot or Audi five cylinder or the growl of a Jag, but the engine has a unique exhaust note that is throaty under acceleration but also disappears into the background when on a long cruise.

The auto box is a highlight with snappy changes when using the paddles on the steering wheel. It’s great fun.

In other words, when it comes to a thoroughly useable hot hatch, this Mercedes AMG A 45 4MATIC is up with the best. If sheer 0-60mph blastability is what matters to you, then you’ll love it. There’s the chance to have some serious performance enjoyment by dialling in the Sport modes or make the most of the roomy interior and decent boot space by selecting Comfort mode and head off with the family on holiday.

Oh, and it looks good in a butch sort of way. Avoid the Aerodynamic Package if you don’t want it to look too shouty, though.

For more info, see Mercedes