the source is 120V DC with an internal resistance of 40 Ohms. the battery's emf is 10V with an internal resistance of 1kOhms. what can i add to the circuit to improve the charging time (faster charge)?
One of the requirements of a "good" rechargeable battery is the full cycle efficiency. If you had a battery that could recycle 100% of the Coulomb charge, but took 14 volts to charge, with only 12 volts out (like a car battery), that is only 86% efficiency. If you charge with 120 volts, and get 10 volts out, that is a very poor efficiency.
Most batteries have a limit on how fast they can be charged, so you need to be careful. The normal way the charge rate is sped up is to increase the voltage. This is how the "smart" chargers for radio control model batteries work, they ramp up to a set amperage by increasing voltage, then periodically shut off the charge current and sample battery voltage.