a ball of mass m , moving with uniform speed, collides elastically with another stationary ball.the incident ball will lose maximum kinetic energy when the mass of the stationary ball is?
the same as the mass of the moving ball.
To find the mass of the stationary ball when the incident ball loses maximum kinetic energy after an elastic collision, we need to consider the concept of conservation of kinetic energy.
During an elastic collision, both momentum and kinetic energy are conserved. Let's denote the mass of the incident ball as m1 and the mass of the stationary ball as m2.
Initially, the total kinetic energy of the system is given by:
KE_initial = 1/2 * m1 * v^2, where v is the initial speed of the incident ball.
After the collision, the final kinetic energy of the system is given by:
KE_final = 1/2 * m1 * v1_final^2 + 1/2 * m2 * v2_final^2, where v1_final and v2_final are the final speeds of the incident and stationary balls, respectively.
In an elastic collision, the total kinetic energy before and after the collision is the same:
KE_initial = KE_final
Therefore:
1/2 * m1 * v^2 = 1/2 * m1 * v1_final^2 + 1/2 * m2 * v2_final^2
Since the incident ball is moving with uniform speed, we can write:
v1_final = v - v2_final, where v is the initial speed of the incident ball.
Substituting this into the equation, we get:
1/2 * m1 * v^2 = 1/2 * m1 * (v - v2_final)^2 + 1/2 * m2 * v2_final^2
Expanding and simplifying the equation, we have:
m1 * v^2 = m1 * v^2 - 2 * m1 * v * v2_final + m1 * v2_final^2 + m2 * v2_final^2
Canceling out common terms, we get:
0 = -2 * m1 * v * v2_final + m1 * v2_final^2 + m2 * v2_final^2
Rearranging the equation, we have:
2 * m1 * v * v2_final = m1 * v2_final^2 + m2 * v2_final^2
Combining like terms, we get:
2 * m1 * v * v2_final = v2_final^2 * (m1 + m2)
Canceling out v2_final from both sides of the equation, we have:
2 * m1 * v = (m1 + m2) * v2_final
Simplifying further, we get:
2 * m1 * v / v2_final = m1 + m2
Finally, isolating m2, we get:
m2 = 2 * m1 * v / v2_final - m1
Therefore, the mass of the stationary ball when the incident ball loses maximum kinetic energy is given by:
m2 = 2 * m1 * v / v2_final - m1