Momentum, Impulse, and Collisions
We demonstrate that the conservation of momentum, \(m \mathbf {v}\) , is a result of integrating Newton’s second law over time. If the integral is zero, that is, if the net force is zero, then we find that \(m \mathbf {v}\) does not change. But how can th
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Momentum, Impulse, and Collisions
Two galaxies collide, leading to millions and millions of stars interacting. Can we say anything general about such a collision? For example, if you know the galaxies velocities after the collision, what can you learn about their velocities before the collision? We have started to study the consequences of Newton’s laws of motion. Our first discovery is the conservation of mechanical energy. For an object subject only to conservative forces, the mechanical energy is conserved. Energy conservation provides a useful tool when solving problems in mechanics: we can relate the velocity and position of an object without having to find the position as a function of time. This is particularly useful when the interactions are complicated, and we have limited knowledge about the forces between objects. Actually, it allows us to reason about the behavior of a system without having force models, as long as we know that the forces are conservative. Conservation of mechanical energy: The conservation of mechanical energy is an example of a conservation law, which we found by integrating Newton’s second law along the path:
t1
Fnet · vdt = ΔK .
(12.1)
t0
For a one-dimensional motion where the net force only depends on the position, x, we get:
x1
x0
Fxnet dx = ΔK =
1 2 1 2 mv − mv . 2 1 2 0
(12.2)
If we could calculate the integral on the left, we could find the velocity as a function of position without finding the complete motion.
© Springer International Publishing Switzerland 2015 A. Malthe-Sørenssen, Elementary Mechanics Using Matlab, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-19587-2_12
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12 Momentum, Impulse, and Collisions
Conservation of momentum: This technique is sometimes called an integration method. To get energy conservation we integrated Newton’s second law over space. But we can also integrate Newton’s second law over time:
t1
t0
F dt = net
t1
madt = mv1 − mv0 .
(12.3)
t0
If the integral on the left is zero, that is, if the net force is zero, then we find that mv does not change. This gives us another conservation law: Conservation of momentum, mv. But how can that be useful? Didn’t we already know from Newton’s first law that if the net external force is zero, the velocity does not change? It turns out that conservation of momentum is not that useful for a single object, but it is very useful for systems consisting of several objects. For systems of several objects, we will demonstrate that the total momentum is conserved if there are no external forces acting on the system. It does not matter what internal forces are acting, the total momentum is conserved at all times as long as there are no external forces. Collisions: Conservation of momentum is particularly useful for collisions. During a collision between two objects, the interactions between the objects can be very complicated, and may consist of both conservative and non-conservative forces, but as long as the objects are not affected by any external forces, their
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