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 25 m/s. The third piece has twice the mass as the other two.
1. What is the speed of the third piece?
2. What is the direction of the third piece?
To solve this problem, we need to analyze the given information and apply the principles of momentum and vectors.
1. To find the speed of the third piece, we can use the principle of conservation of momentum. The total momentum before the explosion is equal to the total momentum after the explosion.
Let's denote the mass of each of the two smaller pieces as m, and the mass of the third piece as 2m.
Before the explosion, the initial momentum is zero because the coconut is at rest.
After the explosion, the momentum of the two smaller pieces can be calculated separately since they fly off perpendicular to each other. Let's denote the speed of each smaller piece as v.
The momentum of the first smaller piece flying off south is calculated as m * v. The momentum of the second smaller piece flying off west is calculated as m * v.
Therefore, the total momentum after the explosion is (m * v) + (m * v) = 2m * v.
Since the total momentum before and after the explosion is the same, we can set it equal to zero:
0 = 2m * v
This equation tells us that either m or v must be equal to zero. However, since the smaller pieces fly off perpendicular to each other at 25 m/s, neither m nor v can be zero.
Therefore, the equation cannot be true, and we conclude that there is an error in the given information. We cannot determine the speed of the third piece with the provided information.
2. Without knowing the speed or direction of the third piece, we cannot determine its direction with certainty. However, we know that the two smaller pieces fly off south and west perpendicular to each other. Therefore, the third piece is likely to be propelled in a direction somewhere between south and west.
To solve this problem, we can 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 label the mass of each of the two smaller pieces as m, and the mass of the third piece as 2m. We also know that the velocity of the two smaller pieces is 25 m/s.
1. Speed of the third piece:
First, let's find the total momentum before the explosion. The momentum of each smaller piece is given by mass times velocity, so the total momentum before the explosion is given by:
Total momentum before = (2m)(0 m/s) + (m)(25 m/s) + (m)(25 m/s) = 50m m/s.
Next, let's determine the total momentum after the explosion. Since only the two smaller pieces are flying off, the total momentum after is given by:
Total momentum after = (m)(25 m/s) + (m)(25 m/s) = 50m m/s.
According to the principle of conservation of momentum:
Total momentum before = Total momentum after
Therefore, we have:
50m m/s = 50m m/s
This means that the velocity of the third piece is 0 m/s. Thus, the speed of the third piece is 0 m/s.
2. Direction of the third piece:
Since the third piece has a speed of 0 m/s, it is not moving. Therefore, its direction is undefined or can be considered stationary.
1. 17.66 m/s
2. 45 degrees Northeast