A firecracker in a coconut blows the coconut into three pieces. Two pieces of equal mass fly off south and west, perpendicular to each other, at 21m/s . The third piece has twice the mass as the other two.
What is the speed of the third piece?
What is the direction of the third piece?
To find the speed of the third piece, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion should be equal to the total momentum after the explosion.
Let's assume the mass of each of the smaller pieces is 'm'. Since there are two pieces with equal mass, their total mass is 2m. The mass of the third piece is given as twice the mass of the smaller pieces, which is 2m.
Before the explosion, the coconut and the firecracker were at rest, so the total initial momentum is zero.
After the explosion, the two smaller pieces fly off in perpendicular directions at the same speed (21 m/s) and have equal mass (m). So let's calculate their momentum.
The momentum of each of the smaller pieces can be calculated by multiplying its mass (m) by its velocity (21 m/s). Since there are two pieces, the total momentum of the smaller pieces is given by:
Momentum of smaller piece #1 = m * 21 m/s
Momentum of smaller piece #2 = m * 21 m/s
Total momentum of the smaller pieces = (m * 21 m/s) + (m * 21 m/s) = 2m * 21 m/s
To calculate the momentum of the third piece, we will use the fact that its mass is twice that of the smaller pieces. So the momentum of the third piece is given by:
Momentum of the third piece = 2m * v3, where v3 is the speed of the third piece.
According to the conservation of momentum, the total momentum before the explosion is equal to the total momentum after the explosion. Therefore:
0 = (2m * 21 m/s) + (2m * v3)
0 = 42m + 2m * v3
Solving this equation for v3, we have:
2m * v3 = -42m
v3 = -42m / (2m)
v3 = -21 m/s
The negative sign indicates that the third piece is moving in the opposite direction of the smaller pieces.
Hence, the speed of the third piece is 21 m/s, and its direction is opposite to the directions of the smaller pieces (i.e., northward or opposite to the direction of south and west).
To solve this problem, we can break it down into two components: the x-component and the y-component.
Let's assign some variables for the given information:
Mass of each small piece: m
Mass of the third piece: 2m
Speed of the small pieces: 21 m/s
First, let's find the total momentum in the x-direction:
Momentum of the small pieces in the x-direction = mass * velocity
Momentum of each small piece in the x-direction = m * (-21 m/s)
The negative sign indicates that the small pieces are moving west.
Since there are two small pieces, the total momentum in the x-direction becomes:
Momentum of the small pieces in the x-direction = 2 * (m * (-21 m/s)) = -42m
Since momentum is conserved in an explosion, the momentum of the third piece must be equal and opposite in the x-direction. Therefore:
Momentum of the third piece in the x-direction = 42m
Next, let's find the total momentum in the y-direction:
Momentum of the small pieces in the y-direction = mass * velocity
Momentum of each small piece in the y-direction = m * (-21 m/s)
The negative sign indicates that the small pieces are moving south.
Since there are two small pieces, the total momentum in the y-direction becomes:
Momentum of the small pieces in the y-direction = 2 * (m * (-21 m/s)) = -42m
Since momentum is conserved in an explosion, the momentum of the third piece must be equal and opposite in the y-direction. Therefore:
Momentum of the third piece in the y-direction = 42m
Now, let's find the magnitude of the momentum of the third piece:
Magnitude of momentum of the third piece = √[(momentum in x-direction)^2 + (momentum in y-direction)^2]
Magnitude of momentum of the third piece = √[ (42m)^2 + (42m)^2 ]
Magnitude of momentum of the third piece = √[ (42^2)(m^2) + (42^2)(m^2) ]
Magnitude of momentum of the third piece = √(2 * 42^2)(m^2)
Magnitude of momentum of the third piece = √(2 * 42^2)(m^2)
Magnitude of momentum of the third piece = √(2 * 42^2)m
Magnitude of momentum of the third piece = (42√2)m
Since momentum is equal to mass multiplied by velocity, we can equate the magnitude of momentum of the third piece to its mass multiplied by its velocity:
(42√2)m = (2m)(velocity of the third piece)
(42√2)m / (2m) = velocity of the third piece
(21√2) = velocity of the third piece
Therefore, the speed of the third piece is 21√2 m/s.
Now, let's find the direction of the third piece. Since it has a negative momentum in the x-direction and a negative momentum in the y-direction, it is moving in the southwest direction.
Therefore, the direction of the third piece is southwest.
I would use conservation of momentum in the S and west direction.
S direction
M*21S+2M*v S=0
so v in the South direction is -10.5
so the same in th eEast direction
v in the East Direction
V=-10.5 E
angle= arctan 10.5/10.5 or NE at velocity 10.5*sqrt2