The answer is Vf 366 ms I just want to know how to get this

A particle of mass m = 1.0 kg, traveling at speed v,-10 m/s, strikes a particle at rest of mass M-4.0 kg, and rebounds in the direction from which it came, with a speed vr. If the amount of heat produced in this collision is 20 J, find ve. (Define all the quantities you introduce in a sketch, and state clearly from what physical laws your initial equations are derived.) 988/

Solution

Initial kinetic energy= 1/2*m*v₀² Since mass M is initially at rest, it does not have K.E initially.

Let speed of M after collision=v_M

Final K.E=1/2*m*v_f²+1/2*M*v_M²

From conservation of energy,

Initial K.E= Final K.E + heat produced

1/2*m*v₀²= 1/2*m*v_f²+1/2*M*v_M² +20---- Equation (1)

From conservation of momentum,

m*v₀=M*v_M -m*v_f --- Equation (2)

From equation 2 we get,

v_M=(10+v_f)/4.

Now substitute this in equation (1),

500=20+0.5*v_f²+1/2*4*((10+v_f)/4)²

Thus solving this quadratic equation, we get v_f=3.66m/s