An unstable atomic nucleus of mass 17.0 ¡Ñ 10¡V27 kg which is at rest disintegrates into three particles. One of the particles, of mass 5.00 ¡Ñ 10¡V27 kg, moves in the y direction with speed 4.50 Mm/s. Another, with mass 8.40 ¡Ñ 10¡V27 kg, moves in the x direction with speed 4.60 Mm/s
i) Calculate the x component of velocity of the third particle
ii) Calculate the y component of velocity of the third particle
To solve this question, we can use the conservation of linear momentum. According to this principle, the total linear momentum before the disintegration must be equal to the total linear momentum after the disintegration.
i) To calculate the x-component of velocity of the third particle (vx3), we can set up the equation for conservation of linear momentum in the x-direction:
(m1 * vx1) + (m2 * vx2) = (m3 * vx3)
Where:
m1 = mass of the first particle = 5.00 × 10^27 kg
vx1 = x-component of velocity of the first particle (unknown)
m2 = mass of the second particle = 8.40 × 10^27 kg
vx2 = x-component of velocity of the second particle = 4.60 Mm/s = 4.60 × 10^6 m/s
m3 = mass of the third particle (unknown)
vx3 = x-component of velocity of the third particle (unknown)
Since the total linear momentum in the x-direction before the disintegration is zero (since the nucleus is at rest), we can rewrite the equation as:
0 = (m3 * vx3) + (m2 * vx2)
Substituting the given values, we get:
0 = (m3 * vx3) + (8.40 × 10^27 kg * 4.60 × 10^6 m/s)
Since the y-component of the third particle's velocity is not given, we assume it to be zero. Therefore, vx3 is the only velocity component of the third particle.
Solving for vx3, we get:
vx3 = -((8.40 × 10^27 kg * 4.60 × 10^6 m/s) / m3)
ii) To calculate the y-component of velocity of the third particle (vy3), we can set up the equation for conservation of linear momentum in the y-direction:
(m1 * vy1) + (m2 * vy2) = (m3 * vy3)
Where:
m1 = mass of the first particle = 5.00 × 10^27 kg
vy1 = y-component of velocity of the first particle = 4.50 Mm/s = 4.50 × 10^6 m/s
m2 = mass of the second particle = 8.40 × 10^27 kg
vy2 = y-component of velocity of the second particle (unknown)
m3 = mass of the third particle (unknown)
vy3 = y-component of velocity of the third particle (unknown)
Again, using the fact that the total linear momentum in the y-direction before the disintegration is zero, we can rewrite the equation as:
0 = (m3 * vy3) + (m2 * vy2)
Substituting the given values, we get:
0 = (m3 * vy3) + (8.40 × 10^27 kg * vy2)
Solving for vy3, we get:
vy3 = -((8.40 × 10^27 kg * vy2) / m3)
To calculate vx3 and vy3, you'll need the mass of the third particle (m3). Unfortunately, the mass of the third particle is not given in the question.
To solve this problem, we can use the law of conservation of momentum. According to this law, the total momentum before the disintegration must be equal to the total momentum after the disintegration.
The initial momentum before disintegration is zero since the nucleus is at rest. Therefore, the total momentum after disintegration is also zero.
Let v_3x be the x-component of velocity of the third particle, and v_3y be the y-component of velocity of the third particle.
i) Since the total momentum after disintegration is zero, the sum of the x-components of momentum of the three particles must be zero. Therefore,
m_2 * v_2x + m_3 * v_3x = 0, where m_2 is the mass of the second particle and v_2x is its x-component of velocity.
Substituting the values, we have
(8.40 × 10^-27 kg) * (4.60 × 10^6 m/s) + m_3 * v_3x = 0
Solving for v_3x, we get
v_3x = -((8.40 × 10^-27 kg) * (4.60 × 10^6 m/s)) / m_3
ii) Similarly, the sum of the y-components of momentum of the three particles must be zero. Therefore,
m_1 * v_1y + m_2 * v_2y + m_3 * v_3y = 0, where m_1 is the mass of the first particle, v_1y is its y-component of velocity, and m_2 and v_2y apply to the second particle.
Since we don't have the values for m_1 and v_1y, we cannot calculate v_3y without more information.
Keep in mind that these calculations assume that there are no external forces acting on the particles during the disintegration.