A block of mass 2 kg is traveling in the positive direction at 3 m/s. Another block of mass 1.5 kg, traveling in the same direction at 4 m/s, collides elastically with the first block. Find the final velocities of the blocks. How much kinetic energy did the system lose

a. The final velocity of the block of mass 2 kg is 3 m/s or 3.86 m/s. The final velocity of the block of mass 1.5 kg is 4 m/s or 2.86 m/s b. The kinetic energy change is 0 J or -12.235 J. Since the collision is elastic, we choose ΔK = 0

Explanation:

From principle of conservation of momentum,

momentum before impact = momentum after impact

Let m₁ = 2 kg, m₂ = 1.5 kg and v₁ = 3 m/s, v₂ = 4 m/s represent the masses and initial velocities of the first and second blocks of mass respectively. Let v₃ and v₄ be the final velocities of the blocks. So,

m₁v₁ + m₂v₂ = m₁v₃ + m₂v₄

(2 × 3 + 1.5 × 4) = 2v₃ + 1.5v₄

6 + 6 = 2v₃ + 1.5v₄

12 = 2v₃ + 1.5v₄

2v₃ + 1.5v₄ = 12 (1)

Since the collision is elastic, kinetic energy is conserved. So

1/2m₁v₁² + 1/2m₂v₂² = 1/2m₁v₃² + 1/2m₂v₄²

1/2 × 2 × 3² +  1/2 × 1.5 × 4² = 1/2 ×2v₃² + 1/2 × 1.5v₄²

9 + 12 = v₃² + 0.75v₄²

21 = v₃² + 0.75v₄²

v₃² + 0.75v₄² = 21  (2)

From (1) v₃ = 6 - 0.75v₄ (3) . Substituting v₃ into (2)

(6 - 0.75v₄)² + 0.75v₄² = 21

36 - 9v₄ + 0.5625v₄² + 0.75v₄² = 21

36 - 9v₄ + 1.3125v₄² - 21 = 0

1.3125v₄² - 9v₄ + 15 = 0

Using the quadratic formula,

v₄ = [-(-9) ± √[(-9)² - 4 × 1.3125 × 15]]/(2 × 1.3125)

= [9 ± √[81 - 78.75]]/2.625

= [9 ± √2.25]/2.625

= [9 ± 1.5]/2.625

= [9 + 1.5]/2.625 or [9 - 1.5]/2.625

= 10.5/2.625 or 7.5/2.625

= 4 m/s or 2.86 m/s

Substititing v₄ into (3)

v₃ = 6 - 0.75v₄ = 6 - 0.75 × 4 = 6 - 3 = 3 m/s

or

v₃ = 6 - 0.75v₄ = 6 - 0.75 × 2.86 = 6 - 2.145 = 3.855 m/s ≅ 3.86 m/s

b. The kinetic energy change ΔK = K₂ - K₁

K₁ = initial kinetic energy of the two blocks =  1/2m₁v₁² + 1/2m₂v₂²

= 1/2 × 2 × 3² +  1/2 × 1.5 × 4² = 9 + 12 = 21 J

K₂ = final kinetic energy of the two blocks = 1/2m₁v₃² + 1/2m₂v₄². Using v = 3 m/s and v = 4 m/s

= 1/2 × 2 × 3² +  1/2 × 1.5 × 4² = 9 + 12 = 21 J.

ΔK = K₂ - K₁ = 21 - 21 = 0

Using v = 3.86 m/s and v = 2.86 m/s

K₂ = 1/2 × 2 × 3.86² +  1/2 × 1.5 × 2.86² = 14.8996 - 6.1347 = 8.7649 J ≅ 8.765 J

ΔK = K₂ - K₁ = 8.765  - 21 = -12.235 J

Since the collision is elastic, we choose ΔK = 0

i need with 1.3.2 investigate cycling o2 and co2 apex

we need to see it

both the fifth and fourteenth amendments of the united states contain a due process clause. while the fifth amendment describes a legal obligation of the federal government to not deprived of life, liberty or property without due process of law, the fourteenth amendment applies the same obligation to the states to guarantee that all of the states provide fair procedures as well.