an object at rest explodes into three pieces of equal mass. one moves east at 20 m/s a second moves southeast at 30 m/s what is the velocity of the third piece??
To find the velocity 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 denote the mass of each piece as m and the velocity of the third piece as v.
Before the explosion, the object is at rest, so its initial momentum is zero. After the explosion, we have the following velocities for the three pieces:
1) The first piece moves east at 20 m/s.
2) The second piece moves southeast at 30 m/s.
To find the velocity of the third piece, we need to determine both its magnitude (speed) and direction. Since the first piece is moving east and the second piece is moving southeast, we can conclude that the third piece must be moving in a direction between east and southeast.
Let's break the velocity of the second piece (30 m/s) into its eastward and southward components. Since it is moving southeast, the angle between its velocity vector and the east direction is 45 degrees.
Using trigonometry, we can find the eastward component of the second piece's velocity:
Eastward component = velocity * cos(angle)
Eastward component = 30 m/s * cos(45°)
Eastward component ≈ 30 m/s * 0.707 ≈ 21.21 m/s
Now, since the total momentum before the explosion is zero, the total momentum after the explosion should also be zero.
Total momentum before explosion = Total momentum after explosion
Since the three pieces have equal masses, we can write the equation as:
(0) = (m * 20 m/s) + (m * 21.21 m/s) + (m * v)
Simplifying the equation:
0 = 20m + 21.21m + mv
Combining like terms:
0 = 41.21m + mv
Now, to find the velocity of the third piece (v), we can isolate the v term:
-41.21m = mv
Dividing both sides by m:
-41.21 = v
Therefore, the velocity of the third piece is approximately -41.21 m/s in the eastward direction. The negative sign indicates that the piece is moving in the opposite direction of the first two pieces.