The consensus on the internet seems to be that radial and normal burns don't change the total energy of the orbit, since you're thrusting perpendicular to your motion. I'm having trouble squaring that with the following scenario:
Imagine a satellite orbiting a body at 4m/s. It then performs a radial impulse burn of 3m/s. It's final speed is the 4m/s prograde, plus the 3m/s radial = 5m/s. It's speed has increased, and since the burn was instant, it hasn't changed its position. Thus, its gravitational energy (a function of position) is the same, and its kinetic energy (a function of speed) has increased. Therefore, different orbital energy.
Where am I going wrong?