Regarding Speed of launched rocket inside the atmosphere I am a 15 yr old, so please forgive my mistakes as I have not entirely read rocket mechanics.
I have been wondering why rockets accelerate to mind-boggling speeds, (mach7-8 or even more)when that must be causing massive drag forces wasting energy.
Why can't the rocket produce little thrust to accelerate upto say mach 1, and then maintain this speed until the rocket is at a high enough altitude where the air resistance is negligible where it can then use the thrusters at total capacity to speed up for space travel or to achieve a stable orbit?
And in this way maybe we can use more efficient but less thrust-producing methods like ion propulsion. (although we may not currently have the technology to do so)
I feel by moving slowly through the atmosphere and then picking up speed in space we can save a lot of much-needed fuel, but none of the rocket launches take place this way so there must be some point I'm missing.
 A: Actually you are correct that drag wastes energy. Actually rockets are designed differently for different purposes.
Those rockets which are designed to send satellites to low earth orbit actually take some help from air drag to slow them down a bit to get them to the parking/ low earth orbit. So they usually go fast initially although this might be different for different agencies.
Those rockets for deep space actually go slower as you predicted, like Saturn V or SLS. Then they accelerate further in deep space. Ion propulsion is actually not appropriate for big bulky rockets and with air drag acting, and are suitable for satellites in deep space.
Further understanding requires calculus and deep math stuff.
Hope it helped you.
A: The reason is that there are actually two types of "drag" acting on the rocket: atmospheric drag and gravity drag (or gravity loss).
Atmospheric drag is what you have mentioned in your question and gets a lot worse the faster you go.
Gravity loss is the effect of the planet pulling the rocket back down against the thrust of the engines. It is a constant* cost until you reach orbit and it reduces the acceleration generated by the engine thrust.
To limit gravity losses you want your engine thrust to be as high as possible so that the proportion of thrust lost to gravity is minimised and the time you spend not in orbit is also minimised.
Take an extreme case where the engines produce an acceleration of 1g. The rocket would hover and not go anywhere, thereby wasting all of its fuel and being 100% inefficient (despite having no aerodynamic drag).
On the other hand, if your engines produce 5g then only 20% is lost to gravity and you still accelerate at 4g.
So, in summary, you need to find a balance such that the rocket's velocity in the atmosphere is not so great that there is excessive aerodynamic force, heating and drag and not so little that you waste all of your fuel fighting gravity.
*Not really constant, but it isn't proportional to velocity.
