an object with mass 10kg is at rest. It explodes into 4 pieces. A peice with mass 3kg travels NORTH at 5m/s, another piece 5kg travels at 30(degrees) NORTH of EAST at 2 m/s, and another piece 1.1kg travels 60(degrees) NORTH of WEST at 7 m/s. Find the mass and path of the fourth piece.
To solve this problem, we'll use the principle of conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.
Let's start by calculating the momentum of each piece after the explosion:
1. Piece with mass 3kg traveling north at 5m/s:
Momentum = mass * velocity = 3kg * 5m/s = 15 kg m/s
2. Piece with mass 5kg traveling at 30 degrees north of east at 2m/s:
To find the momentum in the x and y directions, we need to decompose the velocity vector into its components.
The x-component can be found using cosine:
x-component = velocity * cos(angle) = 2m/s * cos(30 degrees) = 2m/s * √3/2 = √3 m/s
The y-component can be found using sine:
y-component = velocity * sin(angle) = 2m/s * sin(30 degrees) = 2m/s * 1/2 = 1m/s
Momentum = mass * velocity = 5kg * (√3 m/s, 1m/s) = (5√3 kg m/s, 5 kg m/s)
3. Piece with mass 1.1kg traveling at 60 degrees north of west at 7m/s:
To find the momentum in the x and y directions, we need to decompose the velocity vector into its components.
The x-component can be found using sine:
x-component = velocity * sin(angle) = 7m/s * sin(60 degrees) = 7m/s * √3/2 = 7√3/2 m/s
The y-component can be found using cosine:
y-component = velocity * cos(angle) = 7m/s * cos(60 degrees) = 7m/s * 1/2 = 7/2 m/s
Momentum = mass * velocity = 1.1kg * (7√3/2 m/s, 7/2 m/s) = (7.7√3/2 kg m/s, 7.7/2 kg m/s)
Now, let's calculate the total momentum of the system after the explosion by adding the momenta of the three pieces:
Total momentum = (15 kg m/s) + (5√3 kg m/s, 5 kg m/s) + (7.7√3/2 kg m/s, 7.7/2 kg m/s)
Since momentum is a vector, we can separate it into the x and y components:
Total momentum in the x-direction: √3 kg m/s + 5√3/2 kg m/s + 7.7√3/2 kg m/s
Total momentum in the y-direction: 5 kg m/s + 7/2 kg m/s + 7.7/2 kg m/s
Simplifying these expressions:
Total momentum in the x-direction: (√3 + 5√3/2 + 7.7√3/2) kg m/s
Total momentum in the y-direction: (5 + 7/2 + 7.7/2) kg m/s
To find the mass and path of the fourth piece, we know that the total momentum before the explosion was zero since the object was initially at rest. Therefore, the total momentum after the explosion should also be zero.
Since we already found the total momentum in the x and y directions, we can set the equations equal to zero and solve for the unknowns, which represent the mass and the path of the fourth piece. However, we would need additional information or equations to fully determine the values for the fourth mass and path.