You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sofia, with mass 80.0 kg, is given a push and slides eastward. Elena, with mass 55.0 kg, is sent sliding northward. They collide, and after the collision, Sofia is moving at 39.0 ∘ north of east with a speed of 6.40 m/sm/s and Elena is moving at 20.0 ∘ south of east with a speed of 9.40 m/s.
a. What was the speed of each person before the collision?
b. By how much did the total kinetic energy of the two people decrease during the collision?

a)  V_ei = 2.6434 m/s , V_si = 11.05 m/s

b) Loss K.E = 1007.958 J

Explanation:

Given:

- Mass of Sofia Ms = 80 kg

- Mass of Elena Me = 55 kg

- The initial velocity of Sofia = V_si

- The initial velocity of Elena = V_ei

- The final velocity of Sofia V_sf = 6.4 m/s

- The final velocity of Elena V_ef = 9.4 m/s

- The angle sophia makes with east direction towards north θ_1 = 39°

- The angle Elena makes with east direction towards south θ_2 = -20°

Find:

a. What was the speed of each person before the collision?

b. By how much did the total kinetic energy of the two people decrease during the collision?

Solution:

- Since no eternal forces are acting on either Sophia or Elena, then the momentum of both is conserved in all directions.

- Using conservation of momentum in north-south direction. Taking north as positive and south as negative:

                      Me*V_ei = Me*V_ef*sin(θ_2) + Ms*V_sf*sin(θ_1)  

Plug values in:

                      55*V_ei = 55*(9.4)*sin(-20) + 80*6.4*sin(39)

                      55*V_ei = 145.38762

                      V_ei = 2.6434 m/s

- Using conservation of momentum in east-west direction. Taking east as positive and west as negative:

                      Ms*V_si = Me*V_ef*cos(θ_2) + Ms*V_sf*cos(θ_1)

Plug values in:

                      80*V_si = 55*(9.4)*cos(-20) + 80*6.4*cos(39)

                      80*V_si = 883.71981

                      V_si = 11.05 m/s

- The total kinetic energy of both elena and sophia before collision is given by:

                     K.E_i = 0.5*Me*V_ei^2 + 0.5*Ms*V_si^2

                     K.E_i = 0.5*55*2.6434^2 + 0.5*80*11.05^2

                     K.E_i = 5076.258 J

- The total kinetic energy of both elena and sophia after collision is given by:

                     K.E_f = 0.5*Me*V_ef^2 + 0.5*Ms*V_sf^2

                     K.E_f = 0.5*55*9.4^2 + 0.5*80*6.4^2

                     K.E_f = 4068.3 J

- The loss in kinetic energy after collision is given by:

                    Loss K.E = K.E_i - K.E_f

                    Loss K.E = 5076.258 - 4068.3

                    Loss K.E = 1007.958 J