dicusselastic collision in one dimension.Obtain expression for velocities of two bodies after such a collision
In a one-dimensional elastic collision, two objects collide and bounce off each other without any loss of kinetic energy. The conservation of kinetic energy and momentum are the key principles used to obtain expressions for the velocities of the two bodies after the collision.
Consider two objects, labeled object 1 and object 2, with masses m1 and m2, initially moving towards each other with velocities u1 and u2 respectively. After the collision, the velocities of the objects are denoted as v1 and v2.
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be written as:
m1 * u1 + m2 * u2 = m1 * v1 + m2 * v2 ---(1)
Similarly, the total kinetic energy before the collision is equal to the total kinetic energy after the collision, as it is an elastic collision with no energy loss. The kinetic energy is given by the equation:
KE = (1/2) * m * v^2
Applying this to both objects, the expression can be written as:
(1/2) * m1 * u1^2 + (1/2) * m2 * u2^2 = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 ---(2)
Simplifying equation (2) will help us in obtaining the desired expression. Rearranging equation (2) gives:
m1 * u1^2 + m2 * u2^2 = m1 * v1^2 + m2 * v2^2 ---(3)
Now, subtracting equation (3) from equation (1), we get:
m1 * u1 + m2 * u2 - (m1 * u1^2 + m2 * u2^2) = m1 * v1 + m2 * v2 - (m1 * v1^2 + m2 * v2^2)
Rearranging and factoring, we obtain:
m1 * (u1 - v1)(u1 + v1) + m2 * (u2 - v2)(u2 + v2) = 0
Since the collision is elastic, the objects will exchange velocities. Therefore, v1 and v2 can be written in terms of u1 and u2 as follows:
v1 = [(m1 - m2) * u1 + 2 * m2 * u2] / (m1 + m2)
v2 = [(m2 - m1) * u2 + 2 * m1 * u1] / (m1 + m2)
These expressions give the velocities of object 1 (v1) and object 2 (v2) after the elastic collision, based on their initial velocities (u1 and u2) and masses (m1 and m2).