By accident, a large plate is dropped and breaks into three pieces. The pieces fly apart parallel to the floor, with v1 = 2.95 m/s and v2 = 1.70 m/s. As the plate falls, its momentum has only a vertical component, and no component parallel to the floor. After the collision, the component of the total momentum parallel to the floor must remain zero, since the net external force acting on the plate has no component parallel to the floor. Using the data shown in the drawing, find the masses of pieces 1 and 2.

m1 falls at an angle of 25 degrees and m2 falls at an angle of 45 degrees.

There is no way I can do the math, I have no idea how the angles are measured. Break the two given in to ninety degree components, frankly, just establish your own coorinate system. The third piece will have momentum equal and opposite to the sum of the given two.

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision. Let's denote the mass of piece 1 as m1 and the mass of piece 2 as m2.

1. First, let's calculate the momentum for each piece before the collision.
- The momentum of piece 1 before the collision is given by:
p1 = m1 * v1
- The momentum of piece 2 before the collision is given by:
p2 = m2 * v2

2. The total momentum before the collision is the vector sum of the individual momenta. Since the momentum has only a vertical component, we can write the equation for the total momentum before the collision as:
p_total_before = (p1 + p2) * cos(angle1 + angle2)

3. The total momentum after the collision also has only a vertical component and no component parallel to the floor. Therefore, the total momentum after the collision would be zero.
p_total_after = 0

4. Equating the total momentum before and after the collision, we get:
(p1 + p2) * cos(angle1 + angle2) = 0

5. Solving this equation, we can find the mass of piece 2 in terms of the mass of piece 1:
m2 = -(p1 * cos(angle1 + angle2)) / (v2 * cos(angle2))

6. Substitute the given values of v1, v2, angle1, and angle2 into the equation to find the mass of piece 2.

7. Once the mass of piece 2 is known, we can find the mass of piece 1 by using the equation:
m1 = p1 / (v1 * cos(angle1))

8. Substitute the known values of p1, v1, and angle1 into the equation to find the mass of piece 1.

Remember to convert the angles from degrees to radians when using trigonometric functions.

To find the masses of pieces 1 and 2, we can utilize the principles of momentum conservation and vector decomposition.

First, let's consider the components of the momentum for the plate before and after the collision. We are given that the momentum only has a vertical component before the collision and no component parallel to the floor. This means that the vertical component of momentum before the collision is equal to the vertical component of momentum after the collision.

Let's denote the masses of pieces 1 and 2 as m1 and m2, respectively.

Before the collision:
Vertical component of momentum before = m1 * v1 * sin(25°) + m2 * v2 * sin(45°) --- Equation 1
(Note: We use the sine function to account for the angle at which the masses fall.)

After the collision:
Vertical component of momentum after = m1 * v1f * sin(25°) + m2 * v2f * sin(45°) --- Equation 2
(Here, v1f and v2f represent the final velocities of pieces 1 and 2, respectively. Since the plate breaks into three pieces, we consider only the momentum of pieces 1 and 2.)

Since the vertical component of momentum must stay the same (due to no net external force parallel to the floor), we equate Equation 1 and Equation 2.

m1 * v1 * sin(25°) + m2 * v2 * sin(45°) = m1 * v1f * sin(25°) + m2 * v2f * sin(45°)

Now, let's solve for the masses of pieces 1 and 2.

To eliminate the unknown final velocities v1f and v2f, we need another equation. Fortunately, we have one more piece of information: the parallel velocities v1 and v2.

Using the knowledge that the total horizontal momentum before the collision is equal to the total horizontal momentum after the collision (since there are no external forces parallel to the floor), we can write:

Total horizontal momentum before = Total horizontal momentum after

m1 * v1 * cos(25°) + m2 * v2 * cos(45°) = m1 * v1f * cos(25°) + m2 * v2f * cos(45°)

Since the masses m1 and m2 are common to both equations, we can subtract the second equation from the first equation to get rid of the final velocities:

(m1 * v1 * sin(25°) + m2 * v2 * sin(45°)) - (m1 * v1f * sin(25°) + m2 * v2f * sin(45°)) = 0

Rearranging the terms, we have:

m1 * (v1 * sin(25°) - v1f * sin(25°)) + m2 * (v2 * sin(45°) - v2f * sin(45°)) = 0

Now, let's solve for m1:

m1 * (v1 * sin(25°) - v1f * sin(25°)) = - m2 * (v2 * sin(45°) - v2f * sin(45°)) (dividing by the common factor)

m1 = [- m2 * (v2 * sin(45°) - v2f * sin(45°))] / (v1 * sin(25°) - v1f * sin(25°)) --- Equation 3

Similarly, we can solve for m2:

m2 = [- m1 * (v1 * sin(25°) - v1f * sin(25°))] / (v2 * sin(45°) - v2f * sin(45°)) --- Equation 4

To calculate the masses of pieces 1 and 2, you need to plug in the given values for v1, v2, the angles, and the unknown values for v1f and v2f. Solve Equations 3 and 4 simultaneously to find m1 and m2.