shooting object by using a DC motor
To shoot a coin, you need a fair amount of speed. A DC motor will probably accelerate too slowly to achieve the required speed in such a short distance. My advice would be to use something with less travel but more speed.

A solenoid can be used punch something pretty hard. They typically travel less than an inch, but with a good current pulse, will kick pretty hard. If you don't have enough current available, then you can charge up a large capacitor, and discharge it suddenly into the solenoid.

Alternatively, you can mount the coin on a spring which is held back by a little solenoid. When you energise the solenoid, it pulls back a catch which releases the spring which fires the coin. The down side is that you would have to re-set the mechanism each time. (or you could use a motor to pull the catch).

Lastly, as David Kessner suggested, you could have a pair of counter-rotating flywheels. Whacking the box pushes the coin into the wheels, which grip it and throw it skyward. The downside to this is that you need to keep the power on to keep the wheels spinning, which is a bit of a waste.