The velocity of a 1.2 kg hand needs to cause the face temperature to rise 1.8 ^oC when slapped. The kinetic energy of the hand decreases during the process, due to a decrease in velocity from the initial value to zero. At the same time, the internal energy of the affected area increase, due to an increase in the temperature.

Assumptions:

• The hand is brought to a complete stop after the impact and the face does not move.
• No heat is transferred from the affected area to the surroundings. No work is done to or by the system.
• The potential energy change is zero.

System

The energy balance equation:  

      E_in - E_out = Δ E_system
              = (ΔU + ΔKE + ΔPE)_{affected tissue} +
                 (ΔU + ΔKE + ΔPE)_hand

With all these assumptions, the equations can be simplified as:

Plug in the numbers, the solution for velocity v is:

      0 = (mcDT)_{affected tissue} + [m(0 - v²)/2]_hand

This is a very fast hand.