You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass 78.0 kg , is given a push and slides eastward. Abigail, with mass 58.0 kg , is sent sliding northward. They collide, and after the collision Sam is moving at 40.0 ∘ north of east with a speed of 6.20 m/s and Abigail is moving at 21.0 ∘ south of east with a speed of 8.20 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) the speed of Sam was 10.44 m/s and

the speed of Abigail was 2.42 m/s

(b) Decreased by 973.158 J or ΔKE = -973.158 J

Explanation:

Here we have

Initial total momentum = final total momentum

Final total momentum = m₁ 6.2(sin 40 i + cos 40 j) + m₂ 8.2 (-sin 21 i + cos 21 j)

Where m₁ = Mass of Sam = 78.0 kg

m₂ = Mass of Abigail = 58.0 kg

Therefore

m₁ 6.2(sin 40 i + cos 40 j) + m₂ 8.2 (-sin 21 i + cos 21 j)

310.85 i 370.459 j + -170.44 i + 444.01 j

= 140.41 i + 814.5 j

(a) Therefore since the direction of motion of Sam is in the x direction and Abigail in the y direction, we have

we have

78 × v₁ₓ + 58 × 0 = 78 × 4.75 + 58×7.66  

v₁ₓ = 10.44 m/s

Similarly

78×0 + 58×v₂ₙ = 78 × 3.99 - 58× 2.94

v₂ₙ = 140.4/58 = 2.42 m/s

Therefore the speed of Sam was 10.44 m/s and

the speed of Abigail was 2.42 m/s

Their total kinetic energy before collision was

KE initial = 0.5×78×10.44² + 0.5×58×2.42²

= 4422.28 J

Their Kinetic energy after collision is

0.5×78×6.2² + 0.5×58×8.2² = 3449.12

Therefore, their Kinetic energy decreased by

4422.28 J - 3449.12 J = 973.158 J