An unstable nucleus with a mass of 18.0 ✕ 10-27 kg initially at rest disintegrates into three particles. One of the particles, of mass 4.1 ✕ 10-27 kg, moves along the positive y-axis with a speed of 8.0 ✕ 106 m/s. Another particle, of mass 7.7 ✕ 10-27 kg, moves along the positive x-axis with a speed of 4.0 ✕ 106 m/s. (Assume that mass is conserved.)
(a) Determine the third particle's speed.
(b) What is the direction of motion? (from the positive x-axis)
momentum is conserved. In the x direction, find out the x component of the third partical (using momentum)
then, in the y direction, find the component of the third partical.
Now, yuu have the x, y componets of momnetum of the third aprtical. Solve for speed, and direction.
To solve this problem, we will need to apply the law of conservation of momentum and energy. Let's break it down step by step.
Step 1: Find the initial momentum of the system.
Since the initial nucleus is at rest, its initial momentum is zero.
Step 2: Find the final momentum of the system.
The final momentum of the system is equal to the sum of the momenta of the three particles.
a) To find the third particle's speed:
Let the speed of the third particle be v3, and its mass be m3.
Using the conservation of momentum, we can write:
0 = 4.1 ✕ 10^-27 kg * 8.0 ✕ 10^6 m/s + 7.7 ✕ 10^-27 kg * 4.0 ✕ 10^6 m/s + m3 * v3
Simplifying the equation, we get:
m3 * v3 = -4.1 ✕ 10^-27 kg * 8.0 ✕ 10^6 m/s - 7.7 ✕ 10^-27 kg * 4.0 ✕ 10^6 m/s
m3 * v3 = -32.8 ✕10^-21 kg∙m/s - 30.8 ✕ 10^-21 kg∙m/s
m3 * v3 = -63.6 ✕ 10^-21 kg∙m/s
Since the mass of the third particle is not provided, we cannot determine its exact velocity. However, we can calculate the magnitude of the velocity (speed) by dividing the above equation by the mass of the third particle:
v3 = -63.6 ✕ 10^-21 kg∙m/s / m3
Therefore, the third particle's speed v3 is -63.6 ✕ 10^-21 kg∙m/s divided by the mass of the third particle m3.
b) To find the direction of motion:
The direction of motion is in the positive x-axis for one particle and in the positive y-axis for the other particle. The third particle's motion will be in the direction determined by the resultant momentum, which we calculated previously.