To solve this problem, we will use the principles of conservation of momentum and conservation of kinetic energy.
(a) Determine the third particle's speed:
We need to find the speed of the third particle. Let's assume its mass is m and its speed is v.
Conservation of momentum tells us that the total momentum before the disintegration is equal to the total momentum after the disintegration.
Before disintegration (initially at rest) :
Total momentum = 0 kg.m/s
After disintegration:
The first particle, with a mass of 4.1 ✕ 10-27 kg, moves along the positive y-axis with a speed of 8.0 ✕ 106 m/s.
Momentum of the first particle = mass x velocity = (4.1 ✕ 10-27 kg) x (8.0 ✕ 106 m/s)
The second particle, with a mass of 7.7 ✕ 10-27 kg, moves along the positive x-axis with a speed of 4.0 ✕ 106 m/s.
Momentum of the second particle = mass x velocity = (7.7 ✕ 10-27 kg) x (4.0 ✕ 106 m/s)
The third particle, with a mass of m and a speed of v.
Momentum of the third particle = mass x velocity = m x v
According to conservation of momentum:
(4.1 ✕ 10-27 kg) x (8.0 ✕ 106 m/s) + (7.7 ✕ 10-27 kg) x (4.0 ✕ 106 m/s) + m x v = 0 kg.m/s
Simplifying the equation:
(32.8 ✕ 10-21 kg.m/s) + (30.8 ✕ 10-21 kg.m/s) + m x v = 0 kg.m/s
63.6 ✕ 10-21 kg.m/s + m x v = 0 kg.m/s
Since the initial total momentum is zero, the total momentum after disintegration must also be zero. Therefore, we can solve for the third particle's speed:
m x v = -63.6 ✕ 10-21 kg.m/s
v = -63.6 ✕ 10-21 kg.m/s / m
(b) Determine the direction of motion:
To find the direction of the third particle's motion, we need to consider the directions of the other two particles. The first particle moves along the positive y-axis, and the second particle moves along the positive x-axis.
By convention, we take the direction of the positive x-axis as the reference. Since the second particle moves in the positive x-axis direction, the third particle must move in the opposite direction.
Therefore, the direction of motion of the third particle is opposite to the positive x-axis.
Note: It's important to remember that we cannot determine the exact value of m or v without additional information. We can only provide a relationship between them through the conservation of momentum equation.