Abel's solution begins with the principle of conservation of energy – since the particle is frictionless, and thus loses no energy to heat, its kinetic energy at any point is exactly equal to the difference in gravitational potential energy from its starting point. The kinetic energy is 1 2 m v 2 , and since the particle is constrained to move along a curve, its velocity is simply d ℓ ℓ / d t , where ℓ ℓ is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height y 0 to a height y is m g ( y 0 − − y ) , thus: