Question

Arrive at an expression for elastic
collision in one dimension and discuss
various cases. ​

Answer 1

Explanation:

Consider two elastic bodies of masses m1 and m2 moving in a straight line (along positive x direction) on a frictionless horizontal surface as shown in figure given below. Elastic collision in one dimension Mass Initial velocity Final velocity Mass m1 u1 v1 Mass m2 u2 v2 In order to have collision, we assume that the mass m] moves faster than mass m2 i.e., u1 > u2. For elastic collision, the total linear momentum and kinetic energies of the two bodies before and after collision must remain the same. From the law of conservation of linear momentum, Total momentum before collision (pi) = Total momentum after collision (pf) m1 u1 + m2 u2 = m1 v1 + m2 v2 ....…(i) or m1(u1-v1) = m2(v2 - u2)........(ii) Further, This means that for any elastic head on collision, the relative speed of the two elastic bodies after the collision has the same magnitude as before collision but in opposite direction. Further note that this result is independent of mass. Rewriting the above equation for v1 and v2 v1 = v2 + u2 – u2 ...…(vi) Or v2 = u1 + v1 – u2 ....…(vii) To find the final velocities v1 and v2 : Substituting equation (vii) in equation (ii) gives the velocity of as m1 as Similarly, by substituting (vi) in equation (ii) or substituting equation (viii) in equation (vii), we get the final velocity of m2 as Case 1: When bodies has the same mass i.e., m1 = m2, The equations (x) and (xi) show that in one dimensional elastic collision, when two bodies of equal mass collide after the collision their velocities are exchanged. Case 2: When bodies have the same mass i.e., m1 = m2 and second body (usually called target) is at rest (u2 = 0), By substituting m1 = m2 = and u2 = 0 in equations (viii) and equations (ix) we get, from equation (viii) ⇒ v1 = 0 …(xii) from equation (ix) ⇒ v2 = u1 ….. (xiii) Equations (xii) and (xiii) show that when the first body comes to rest the second body moves with the initial velocity of the first body. Case 3: The first body is very much lighter than the second body (m1<