Can a Mach 2 missile attain a speed of Mach 5 if launched from an aircraft travelling at Mach 3?

Let's make things easier, suppose you're riding an e-bike and suddenly your new iPhone X dropped down from your pocket; what will happen? do you think your beautiful iPhone X will fly with you by same speed as you riding?!

The answer is yes if there is no air or *air resistance* and gravity.

So, back to your question, assume that your iPhone X can fly with 10 m/s and you're riding with 40 m/s. So now, according to your question your iPhone X will fly with 50 m/s (10+40); which means you will not be able to return back your 1000$.

When a missile leaves an aircraft in a specific speed, a momentum transfer occurs between the two objects in accordance with the momentum conservation law. This specifically affects the aircraft speed which here, regarding your question, we may leave this part behind.

By the way, the missile firing from the aircraft is simply treated as a projectile with an initiating absolute speed. This absolute speed is obtained from the addition of relative speeds generally in vector forms. Therefore, assuming an amplifying analogous direction, the missile simply takes the initiating speed of 5 Mach; and then it is treated as a projectile where the various forces are involved forming a certain trajectory for the missile.

Practically, the missiles are designed to be launched at certain speeds in the requirements. Higher speeds will generate a greater drag as explained by James Garry. It will also increase the heat through friction that the missile will certainly not be able to sustain. The vibrations generated by the shock-waves and controls will certainly damage the missile and its control. When manoeuvring in the last seconds to hit the target (it depends on the type of missile) the acceleration will also be too high. Finally, you need to consider the propulsion system. It will only be able to produce a certain amount of thrust (especially for a rocket) and will not be able to equal the drag. The ramjet will be designed for a certain range of Mach cones, and will not be able to work at greater speed (e.g. the combustion will not have time to occur in a higher speed flow). The intake of a jet engine (I doubt missile use them when designed for supersonic flight) will not be able to slow down the flow correctly, and shocks impinging the blades would considerably reduce their efficiency if not destroy them.

After the missile is launched from the aircraft at Mach 3, if the thrust produced by the missile is greater than the drag experienced at Mach 3, then yes, it can slowly accelerate towards Mach 5 till burnout or till fuel is available.

Good question..its like projectile velocity simple

As long as velocities are smaller than the velocity of light in vacuum, they can be simply added. Like cricket bowler adding his velocity by running and rotating his hand. You can launch a missile at 2 Mach from a place which is moving with velocity 3 Mach, then it can have velocity 5 Mach. Mach is just a number.

A "Mach 2" missile does not mean much. It must be further specified.

However, let us assume that the "Mach 2" missile uses a rocket motor and, after launch, its trajectory is, roughly, a straight line. At Mach 2 the thrust of the "Mach 2" missile motor equals the drag of the missile. At launch, at Mach 3, the drag is greater than the thrust. Therefore, after launch the "Mach 2" missile is decelerating from Mach 3 until it reaches Mach 2 where, probably, it remains for a short period of time. 

Actually, if the "Mach 2" is an IR homing, proximity fuse missile, due to its deceleration, it may even shot down the Mach 3 flying aircraft from which it was launched.  

A mach 2 missile, is a missile that can stand aerodynamic drag caused by air flying at mach 2 speed. It can also safely withstand shocks upto mach 2 whilst flying in its designed velocity. In other words, a mach 2 missile is a missile that "can achieve speeds upto" upto 2000m/s in air. Not saying it can immediately accelerate to this speed, but it "can" withstand the pressure and reach such speeds.

Thus releasing such a missile at a speed of mach 3, will immediately put an unbearable amount of air-friction, mach shock, and form drag on the projectile; thus greatly damaging the missile if not destroying it. because it is "not design to withstand such dynamic force".

It is common for planes to fire missiles mid-air, but they always fire missiles that are designed to achieve speeds higher than the plane deploying them; i.e a plane flying at mach 2, can fir a missile capable of reaching mach 3. This is not to say the missile will fly at mach (2+3=5); in fact this means the missile being fire at mach 2 can accelerate and reach speeds of mach 3 safely.