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.
V1 M1 fall at an angle of 25 degrees and v2 m2 fall at a degree of 45
m3 is 1.30 kg at 3.07 m/2
To find the masses of pieces 1 and 2, we can start by analyzing the problem using the conservation of momentum. Considering that the component of the total momentum parallel to the floor should remain zero, we can determine the velocities of the two pieces after the collision.
Given:
v1 = 2.95 m/s (velocity of piece 1)
v2 = 1.70 m/s (velocity of piece 2)
m3 = 1.30 kg (mass of piece 3)
v3 = 3.07 m/s (velocity of piece 3)
The component of momentum parallel to the floor before the collision is given by:
p_parallel = m1 * v1 * cos(25°) + m2 * v2 * cos(45°) + m3 * v3 * cos(180°)
Since the component of total momentum parallel to the floor must remain zero, this equation becomes:
0 = m1 * v1 * cos(25°) + m2 * v2 * cos(45°) - m3 * v3 * cos(180°)
Now, we need to express the cosines of the angles in terms of sines:
cos(25°) = sin(90° - 25°) = sin(65°)
cos(45°) = sin(90° - 45°) = sin(45°)
cos(180°) = -1 (cosine of 180° is -1)
Substituting these values, we get:
0 = m1 * v1 * sin(65°) + m2 * v2 * sin(45°) - m3 * v3
Now, let's substitute the given values to solve for m1 and m2.
Substituting v1 = 2.95 m/s, v2 = 1.70 m/s, m3 = 1.30 kg, and v3 = 3.07 m/s, the equation becomes:
0 = m1 * 2.95 * sin(65°) + m2 * 1.70 * sin(45°) - 1.30 * 3.07
Next, we need to solve for the masses. Let's solve for m1 first.
m1 * 2.95 * sin(65°) = 1.30 * 3.07 - m2 * 1.70 * sin(45°)
m1 = (1.30 * 3.07 - m2 * 1.70 * sin(45°)) / (2.95 * sin(65°))
Now we can substitute this expression for m1 into the main equation:
0 = (1.30 * 3.07 - m2 * 1.70 * sin(45°)) / (2.95 * sin(65°)) * 2.95 * sin(65°) + m2 * 1.70 * sin(45°) - 1.30 * 3.07
Simplifying the equation by canceling out common terms and solving for m2:
0 = 1.30 * 3.07 - m2 * 1.70 * sin(45°) + m2 * 1.70 * sin(45°) - 1.30 * 3.07
0 = 3.991 - 3.991
0 = 0
The equation simplifies to 0 = 0, which means that m2 can have any value. Since m2 can have any value, we cannot determine the mass of piece 2. However, we can still solve for the mass of piece 1 using the previously derived expression.
m1 = (1.30 * 3.07 - m2 * 1.70 * sin(45°)) / (2.95 * sin(65°))
Now, you can substitute the value of m2 that you want to consider into this expression to find the mass of piece 1.