Kinetic Energy and Momentum - Ek = p2/2m

Kinetic energy can be written in terms of momentum. Starting from E_k = ½mv² and p = mv, so v = p/m:E_k = ½m(p/m)² = p²/2mRearranged, p = √(2mE_k). This form is useful when momentum is known (for example from conservation of momentum, or from the radius of a particle track) and the kinetic energy is needed, or the other way round. It applies at non-relativistic speeds.An important result: for objects with the same momentum, E_k is inversely proportional to mass, so the lighter object has more kinetic energy. When an object at rest explodes into two fragments, they have equal and opposite momenta, so the kinetic energy is shared in the inverse ratio of their masses.Worked example 1: a 3.0 kg trolley has momentum 6.0 kg m s^-1.
E_k = 6.0² / (2 × 3.0) = 36 / 6.0 = 6.0 J
(Check: v = 2.0 m s^-1, and ½ × 3.0 × 2.0² = 6.0 J.)Worked example 2: a stationary 5.0 kg object explodes into a 1.0 kg and a 4.0 kg fragment, releasing 50 J of kinetic energy. The momenta are equal in size, so E_k ∝ 1/m and the energies are in the ratio 4 : 1. The 1.0 kg fragment gets 4/5 × 50 = 40 J and the 4.0 kg fragment gets 10 J.