Beetroot itself doesn't make you run faster but when you eat beetroot or drink beetroot juice it can improve your energy level so when your energy level is improved you'll run faster.
So by eating good nutritious food such as beetroot or drinking beetroot juice and other foods that give you energy then yes you could likely run faster.
There's no guarantee though that you'll start running faster when drinking beetroot juice or eating beetroot though.
Still beetroot is very healthy for you anyway and you should see improved energy levels and it can also detox your body.