Answer: 1) 0. 2) 4.2 Kg. 3) 15.4 m/s 4) 12.9 m/s 5) 0. 6) 3.62 KJ.
Explanation:
1) Assuming that no external forces act during the collision, total momentum must be conserved. As initially the total mass was at rest, so initial momentum is zero, final momentum of all the system must be 0 also.
2) After the explosion, as mass must be conserved also, the sum of the masses of the three pieces must be equal to the original total mass, so we can write the following:
m₁ + m₂ + m₃ = M = 14.6 Kg = 4.9 Kg + 5.5 Kg + m₃
Solving for m₃, we have:
m₃ = 14.6 Kg - 4.9 Kg -5.5 Kg = 4.2 Kg.
3) and 4)
As momentum is a vector, if it is magnitude must be 0, this means that all his components must be 0 too.
So, we can write two equations, one for the x-component, and other for the y-component, as follows:
pₓ = m₁. v₁ₓ + m₂.v₂ₓ + m₃.v₃ₓ = 0
py = m₁.v₁y + m₂. v₂y + m₃. v₃y =0
Replacing by the values, and solving for v₃ₓ and v₃y, we get:
v₃ₓ = 15.4 m/s
v₃y = 12.9 m/s
v = √(15.4)²+(12.9)² = 20.1 m/s
5) As the center of mass must move as if all the mass were concentrated in this point, and we know that the total momentum must be 0, this tells us that the magnitude of the velocity of the center of mass must be 0 too.
6) As initial kinetic energy is 0, as the mass was at rest, the increase in the kinetic energy is obtained simply adding the kinetic energy of every piece of mass gained after explosion, as follows:
K = K₁ + K₂ + K₃ = 1/2 (m₁ . v₁² + m₂.v₂² + m₃.v₃²)
Replacing by the values, we get:
K= 3.62 KJ
The magnitude of the final momentum of the system (all three pieces) is equal to 0 kg-m/s in accordance with the law of conservation of momentum.
Since there's no external force acting on the object during the collision, its total momentum must be conserved. Also, this would make the object to be at an initial state of rest and have a zero initial momentum.
Pi = Pf
0 = Pf
Therefore, the magnitude of the final momentum of the system (all three pieces) is equal to 0 kg-m/s.
By applying the law of conservation of mass, the total mass of the system is given by:
Mt = M₁ + M₂ + M₃
M₃ = Mt - M₁ - M₂
M₃ = 14.6 - 4.9 - 5.5
M₃ = 4.2 kg.
For the x-component of the velocity of the third piece, we have:
P₁ₓ + P₂ₓ + P₃ₓ = 0
P₃ₓ = -P₁ₓ - P₂ₓ
Note: P₁ₓ = -m₁v₁ₓcosθ and P₂ₓ = m₂v₂ₓsinθ
m₃v₃ₓ = m₁v₁ₓcosθ - m₂v₂ₓsinθ
v₃ₓ = [m₁v₁ₓcosθ₁ - m₂v₂ₓsinθ₂]/m₃
v₃ₓ = [(4.9×25.6×cos22) - (5.5×20.6×sin27)]/4.2
v₃ₓ = [116.31 - 51.4]/4.2
v₃ₓ = 15.5 m/s.
For the y-component of the velocity of the third piece, we have:
v₃y = [m₂v₂ycosθ₂ - m₁v₁ysinθ₁]/m₃
v₃y = [(5.5×20.6×cos27) - (4.9×25.6×sin22)]/4.2
v₃y = [100.95 - 46.99]/4.2
v₃y = 12.9 m/s.
Also, the velocity of the third piece is:
V₃ = √(15.5)² + (12.9)²
V₃ = 20.1 m/s.
After the collision, the magnitude of the velocity of the center of mass of the pieces would start, stay and end at rest because the total momentum must be equal to zero (0). Thus, it's velocity is equal to zero (0).
During the explosion, the increase in kinetic energy is given by:
KEi + KEf = 0
KEf = KEf₁ + KEf₂ + KEf₃
KEf = 1/2 × (m₁v₁² + m₂v₂² + m₃v₃²)
KEf = 1/2 × (4.9×25.6² + 5.5×20.6² + 4.2×20.1²)
KEf = 1/2 × (1,696.842 + 2,333.98 + 3,211.264)
KEf = 3621 Joules.
Read more on momentum here: brainly.com/question/15517471
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