conservation linear momentum
Answers
Explanation:
What is the principle of conservation of momentum?
In physics, the term conservation refers to something which doesn't change. This means that the variable in an equation which represents a conserved quantity is constant over time. It has the same value both before and after an event.
There are many conserved quantities in physics. They are often remarkably useful for making predictions in what would otherwise be very complicated situations. In mechanics, there are three fundamental quantities which are conserved. These are momentum, energy, and angular momentum. Conservation of momentum is mostly used for describing collisions between objects.
Just as with the other conservation principles, there is a catch: conservation of momentum applies only to an isolated system of objects. In this case an isolated system is one that is not acted on by force external to the system—i.e., there is no external impulse. What this means in the practical example of a collision between two objects is that we need to include both objects and anything else that applies a force to any of the objects for any length of time in the system.
If the subscripts iii and fff denote the initial and final momenta of objects in a system, then the principle of conservation of momentum says
\mathbf{p}_\mathrm{1i} + \mathbf{p}_\mathrm{2i} + \ldots = \mathbf{p}_\mathrm{1f} + \mathbf{p}_\mathrm{2f} + \ldotsp
1i
+p
2i
+…=p
1f
+p
2f
+…p, start subscript, 1, i, end subscript, plus, p, start subscript, 2, i, end subscript, plus, dots, equals, p, start subscript, 1, f, end subscript, plus, p, start subscript, 2, f, end subscript, plus, dots
Why is momentum conserved?
Conservation of momentum is actually a direct consequence of Newton's third law.
Consider a collision between two objects, object A and object B. When the two objects collide, there is a force on A due to B—F_\mathrm{AB}F
AB
F, start subscript, A, B, end subscript—but because of Newton's third law, there is an equal force in the opposite direction, on B due to A—F_\mathrm{BA}F
BA
F, start subscript, B, A, end subscript.
F_\mathrm{AB} = - F_\mathrm{BA}F
AB
=−F
BA
F, start subscript, A, B, end subscript, equals, minus, F, start subscript, B, A, end subscript
The forces act between the objects when they are in contact. The length of time for which the objects are in contact—t_\mathrm{AB}t
AB
t, start subscript, A, B, end subscript and t_\mathrm{BA}t
BA
t, start subscript, B, A, end subscript—depends on the specifics of the situation. For example, it would be longer for two squishy balls than for two billiard balls. However, the time must be equal for both balls.
t_\mathrm{AB} = t_\mathrm{BA}t
AB
=t
BA
t, start subscript, A, B, end subscript, equals, t, start subscript, B, A, end subscript
Consequently, the impulse experienced by objects A and B must be equal in magnitude and opposite in direction.
F_\mathrm{AB}\cdot t_\mathrm{AB} = – F_\mathrm{BA}\cdot t_\mathrm{BA}F
AB
⋅t
AB
=–F
BA
⋅t
BA
F, start subscript, A, B, end subscript, dot, t, start subscript, A, B, end subscript, equals, –, F, start subscript, B, A, end subscript, dot, t, start subscript, B, A, end subscript
If we recall that impulse is equivalent to change in momentum, it follows that the change in momenta of the objects is equal but in the opposite directions. This can be equivalently expressed as the sum of the change in momenta being zero.
\begin{aligned}m_\mathrm{A} \cdot \Delta v_\mathrm{A} &= -m_\mathrm{B} \cdot \Delta v_\mathrm{B} \\ m_\mathrm{A} \cdot \Delta v_\mathrm{A} + m_\mathrm{B} \cdot \Delta v_\mathrm{B} &= 0\end{aligned}
m
A
⋅Δv
A
m
A
⋅Δv
A
+m
B
⋅Δv
B
=−m
B
⋅Δv
B
=0