How Does Momentum Conservation Apply in Nuclear Disintegration?

[riseofphoenix]

Conservation of Momentum?

An unstable nucleus of mass 2.7 x 10^-26 kg, initially at rest at the origin of a coordinate system, disintegrates into three particles. One particle, having a mass of m₁ = 1.0 x 10^-26 kg, moves in the positive y-direction with speed v₁ = 4.8 x 10⁶ m/s. Another particle, of mass m₂ = 1.2 x 10^-26 kg, moves in the positive x-direction with speed v₂ = 3.6 x 10⁶ m/s. Find the magnitude and direction of the velocity of the third particle.

Magnitude: ______ m/s
Direction: ______ ° counterclockwise from the +x-axis

This is what I did...

1) M = m₁ + m₂ + m₃

(2.7 x 10^-26) = (1.0 x 10^-26) + (1.2 x 10^-26) + m₃
(2.7 x 10^-26) - (1.0 x 10^-26) - (1.2 x 10^-26) = m₃
5.0 x 10^-27 = m₃

2) Determine your systems

1st particle:
m₁ = 1.0 x 10^-25 kg
v_initial = 0 m/s
v_1final = 4.8 x 10⁶ m/s

2nd particle:
m₂ = 1.2 x 10^-26 kg
v_initial = 0 m/s
v_2final = 3.6 x 10⁶ m/s

3rd particle:
m₃ = 5.0 x 10^-27 kg
V_initial = 0 m/s
V_3final = ? m/s

3) Conservation of momentum equation

p_initial = p_final

m₁v_initial + m₂v_initial + m₃v_initial = m₁v_1final + m₂v_2final + m₃v_3final
(m₁)(0) + (m₂)(0) + (m₃)(0) = (1.0 x 10^-25)(4.8 x 10⁶) + (1.2 x 10^-26)(3.6 x 10⁶) + (5.0 x 10^-27)(v_3final)
0 = (4.8 x 10^-20) + (4.32 x 10^-20) + (5.0 x 10^-27)(v_3final)
-(9.12 x 10^-20) = (5.0 x 10^-27)(v_3final)
-18240000 = v_3final
-1.82e+07 = v_3final

"INCORRECT. CORRECT ANSWER IS: 1.27e+07"

I don't understand! How did they get 1.27e+07? Help!

 

[SammyS]

An unstable nucleus of mass 2.7 x 10^-26 kg, initially at rest at the origin of a coordinate system, disintegrates into three particles. One particle, having a mass of m₁ = 1.0 x 10^-26 kg, moves in the positive y-direction with speed v₁ = 4.8 x 10⁶ m/s. Another particle, of mass m₂ = 1.2 x 10^-26 kg, moves in the positive x-direction with speed v₂ = 3.6 x 10⁶ m/s. Find the magnitude and direction of the velocity of the third particle.

Magnitude: ______ m/s
Direction: ______ ° counterclockwise from the +x-axis

This is what I did...

1) M = m₁ + m₂ + m₃

(2.7 x 10^-26) = (1.0 x 10^-26) + (1.2 x 10^-26) + m₃
(2.7 x 10^-26) - (1.0 x 10^-26) - (1.2 x 10^-26) = m₃
5.0 x 10^-27 = m₃

2) Determine your systems

1st particle:
m₁ = 1.0 x 10^-25 kg
v_initial = 0 m/s
v_1final = 4.8 x 10⁶ m/s

2nd particle:
m₂ = 1.2 x 10^-26 kg
v_initial = 0 m/s
v_2final = 3.6 x 10⁶ m/s

3rd particle:
m₃ = 5.0 x 10^-27 kg
V_initial = 0 m/s
V_3final = ? m/s

3) Conservation of momentum equation

p_initial = p_final

m₁v_initial + m₂v_initial + m₃v_initial = m₁v_1final + m₂v_2final + m₃v_3final
(m₁)(0) + (m₂)(0) + (m₃)(0) = (1.0 x 10^-25)(4.8 x 10⁶) + (1.2 x 10^-26)(3.6 x 10⁶) + (5.0 x 10^-27)(v_3final)
0 = (4.8 x 10^-20) + (4.32 x 10^-20) + (5.0 x 10^-27)(v_3final)
-(9.12 x 10^-20) = (5.0 x 10^-27)(v_3final)
-18240000 = v_3final
-1.82e+07 = v_3final

"INCORRECT. CORRECT ANSWER IS: 1.27e+07"

I don't understand! How did they get 1.27e+07? Help!

Velocity and momentum are vector quantities.

You need to keep track of vector components.

 

[riseofphoenix]

Velocity and momentum are vector quantities.

You need to keep track of vector components.

I STILL get the same answer though...

0 = (4.8 x 10^-20) + (4.32 x 10^-20) + (5.0 x 10^-27)(v_3final)

x component


0 = [STRIKE](4.8 x 10^-20)[/STRIKE] + (4.32 x 10^-20) + (5.0 x 10^-27)(v_3final)
0 = (4.32 x 10^-20) + (5.0 x 10^-27)(v_3final)
-(4.32 x 10^-20) = (5.0 x 10^-27)(v_3final)
-(4.32 x 10^-20)/(5.0 x 10^-27) = v_3final
-8.64 x 10⁶= v_3final in the x direction

y component


0 = (4.8 x 10^-20) + [STRIKE](4.32 x 10^-20)[/STRIKE] + (5.0 x 10^-27)(v_3final)
0 = (4.8 x 10^-20) + (5.0 x 10^-27)(v_3final)
-(4.8 x 10^-20) = (5.0 x 10^-27)(v_3final)
-(4.8 x 10^-20)/(5.0 x 10^-27) = v_3final
-9.6 x 10⁶ = v_3final in the y direction

Add vectors

-8.64 x 10⁶ - 9.6 x 10⁶ = -18240000 = -1.82 x 10^7 m/s

-____-

 

[Doc Al]

Add vectors

-8.64 x 10⁶ - 9.6 x 10⁶ = -18240000 = -1.82 x 10^7 m/s

How do you find the magnitude of a vector given its components? (You don't just add the components!)

 

[riseofphoenix]

How do you find the magnitude of a vector given its components? (You don't just add the components!)

But I just did!

 

[Nytik]

Yeah, he's saying it's wrong to just add the components.

Consider the vector and it's components as a triangle (I assume you're familiar with this representation). You're trying to the find the length of the hypotenuse.