Lets say that a 777 climbing out at 1500fpm (sounds a little sluggish to me) is doing 180kts.
That means by the time its climbed from 50 feet to 1500, its flown about 3 miles horizontally - probably nearly twice the runway length
So long as the light a/c is above 1500ft and routing at 90(ish) degrees to the runway, theres not likely to be a confliction.
Even at 3000fpm, which strikes me as being more likely, there is still plently of room - it would be a magnificent sight from the light a/c!