Three guns are aimed at the center of a circle,and each fires a bullet simultaneously. The directions in which they fire are 120 degrees apart. Two of the bullets have the same mass of 4.50*10^-3 kg and the same speed of 324 m/s. The other bullet has an unknown mass and a speed of 575 m/s. The bullets collide at the center and mash into a stationary lump. What is the unknown mass?

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 consider the bullets individually and calculate their momenta before the collision.

Momentum (p) is calculated using the formula:
p = mass * velocity

For the two bullets with the known mass and speed:
p1 = (4.50 * 10^-3 kg) * (324 m/s)
p2 = (4.50 * 10^-3 kg) * (324 m/s)

Since the direction of motion is the same for all bullets, we do not need to worry about the vector nature of momentum.

Now, let's calculate the total momentum before the collision:
total momentum before = p1 + p2 + p3

We know the total momentum after the collision is zero because the bullets collide and come to a halt. Therefore, we can set up the equation:
total momentum before = 0

Substituting the values, we get:
(p1 + p2) + p3 = 0

Since p1 and p2 are equal, we can simplify the equation to:
2p1 + p3 = 0

Now, let's substitute the values of p1 and p3, and solve for the unknown mass (m3):

2 * (4.50 * 10^-3 kg * 324 m/s) + m3 * (575 m/s) = 0

Simplifying the equation, we get:
0.02916 kg·m/s + 575 m/s * m3 = 0

Now, isolate the unknown mass (m3):
m3 = -0.02916 kg·m/s / 575 m/s
= -0.00005069 kg

Note that the negative sign indicates that the direction of m3 is opposite to the other bullets.

Therefore, the unknown mass of the third bullet is approximately -0.00005069 kg.

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