One Dimensional Elastic Collision - Velcoities After Collision

One Dimensional Elastic Collision – Derivation For Velocities Of Two Bodies After Collision Consider two bodies, say spheres, moving in the same direction along a straight line and have an elastic collision. Mass of the first body, Mass of the second body Velocity of the first body before collision Velocity of the second body before collision Velocity of the first body after collision Velocity of the second body after collision Collision occurs only when the velocity of the first body is greater than that of the other, which is . According to the law of conservation of momentum Rearranging the terms in equation 1, we get: According to the law of conservation of energy, Kinetic Energy of the two bodies before collision = Kinetic Energy of the two bodies after collision Simplifying equation 3, we get Equations 2 and 4 constitute the relation between masses and velocities of the bodies before and after collision. To get the relation between the velocities of the bodies before and after collision alone we divide equation 4 by equation 2 and simplify. Substituting in equation 1 and simplifying: From equation 5, we get: Substituting in equation 1 and simplifying: Special Cases: Case 1: If the mass of the two spheres is equal ⇒The two spheres exchange their velocities after collision. Case 2: If the mass of the two spheres is equal and second sphere is at rest before collision ⇒The first sphere comes to rest and the second sphere moves with the velocity of the first one before collision. Case 3: If the mass of the first sphere is much greater than the second sphere and the second sphere at rest ⇒The first sphere continues to move with the same velocity and the second sphere moves with a velocity equal to double that of the first sphere before collision. Case 4: If the mass of the first sphere is much lesser than the second sphere and the second sphere at rest ⇒The first sphere bounces back with the same velocity and the second sphere continues to be at rest.