How often do police take your picture just because they like your car? Not very often, presumably. In which case, try driving the latest electric sportscar from Tesla Motors, the Roadster Sport.

Being the first British newspaper journalist behind the wheel of this £87,000 superstar new model – one that has been Anglicised with a right-hand drive – is a strange experience. Driving it around London, people literally stop, stare, gawp and nudge their friends and children. The jaws of two men drop simultaneously; I’m not sure if they are more impressed by the car or horrified to see a woman driving it. And Dave, a community support police officer in central London, can’t resist taking a photograph. “My brother would kill me if I didn’t,” he says, peering inside afterwards . A few minutes later when I ask a police officer for directions, his eyes light up. “Is that that new electric car?” he asks, as his partner rolls his eyes. I’ve never experienced anything like it.

But what about the driving? First of all, you’re incredibly low down on the road (let’s skip quickly over the business of clambering in and out – not graceful, to say the least) and at moments on the London roads I feel like a weeny unprotected child, in between all the double decker buses and coaches.

Secondly, it’s surprisingly heavy – that’s the weight of the bank of lithium-ion batteries that keeps it moving – and like many sports cars it doesn’t have power-steering. The power behind its famous 0-60mph in 3.7 seconds is not instantly obvious, the weight making it slightly less nippy than you would expect in the traffic. The braking (regenerative obviously) is joltingly powerful – I nearly put the Guardian’s camerawoman through the window several times.

It is an automatic, which takes a little getting used to, but is then heaven. And there’s a neat little display on the dashboard which shows how much current you’re using – two amps while sitting in traffic, and up to 68 when driving at high speed. The dashboard is actually a little over-complicated, and the speed dial is positioned awkwardly behind the steering wheel so you can’t see it unless you duck a little (or maybe I should have been taller.)

However, the place where the Tesla finally stops feeling strange and starts to feel extraordinary is – as you might expect – the fast lane of the motorway. Without a private track we can’t go from a standing start to try out the acceleration experience that nearly caused Jeremy Clarkson to swallow his own dentures on Top Gear. But I went for a spin on the M4 and it was instantly powerful. One moment we are doing 55mph and the next we were doing 70. Other cars just drop away like falling fruit.

But adrenalin kicks aside, why should we care about the Tesla? I would argue that it’s one of the most important cars ever made. Back in 2006 the idea of the electric car was dying – see Chris Paine’s documentary Who Killed the Electric Car? – as the giant car companies dragged their feet and then either brought out models with restricted availability, dumped them or just threw up their hands and said “it’s impossible”. Nickel-metal hydride batteries could not provide the range that was needed and there didn’t seem to be much else available.

And then, like Sir Galahad in a sunlit clearing, the Tesla appeared. Unlike the unattractive and slow city cars that had made up most electric history, it was slinky, bright red, desirable and capable of sportscar-worthy performance off a bank of lithium-ion batteries (the batteries that lap-top computers use). Robert Lutz, vice-chairman of General Motors, has been quoted as saying that “all the geniuses here at General Motors kept saying lithium-ion technology is 10 years away, and Toyota agreed with us – and boom, along comes Tesla. So I said, ‘How come some tiny little California startup, run by guys who know nothing about the car business, can do this, and we can’t?’ That was the crowbar that helped break up the log jam.”

In the years since the log jam appears to have nearly disappeared, with Renault, Nissan, BMW, Mitsubishi and GM itself all taking the electric car seriously these days. The Leaf, the i-MiEV, and the electric Mini are the new generation of EVs which are going to be appearing all over Europe this year and next; they’re all good to drive, they’re modelled like a normal petrol car – rather than the Marmite love-it-or-hate design of the G-Wiz – and the car manufacturers have worked out that if they lease you the expensive battery instead of selling it with the car, then they’ll be priced like, well, any other car.

But for now the Tesla Roadster is very much not like any other car. Just ask a policeman.