# Kinetic energy of a system appears to increase

An object of mass $m$ moves at a speed of $u$ m/s to the right.

It then splits into two objects of equal mass, one of which begins moving at a constant speed, an angle of $\theta$ above the initial motion of the object, and the other begins moving at a constant speed, at an angle of $90^o - \theta$ below the initial motion of the object.

Assuming that momentum is conserved, I found that the initial kinetic energy of the system is $\frac{1}{2}mu^2$, but the final kinetic energy of the system is $mu^2$. Have I made a mistake or can the total kinetic energy of a system increase during a collision?

• Can I know the source of this problem? Jul 7, 2018 at 7:28
• during a collision You are not describing a collision - no objects collide. Is there something missing from the problem ? Jul 7, 2018 at 7:32
• @StephenG Fragmentation is usually included within the topic of the mechanics of collisions. If time is reversed, this scenario is an inelastic collision. Jul 7, 2018 at 7:39
• @sammygerbil I'm actually wondering if the OP has left out an object or the fragmentation is due to a collision with something not mentioned. Jul 7, 2018 at 7:42
• @StephenG I don't get that impression. An object of mass $m$ ... then splits into 2 objects of equal mass ... assuming that momentum is conserved ... is consistent with an isolated fragmentation. Both mass and momentum are conserved. Jul 7, 2018 at 7:55

## 3 Answers

Yes, you are describing a situation in which kinetic energy is not conserved. One way this could happen is if a spring connecting the two particles is released, pushing one particle away from the other. The potential energy of the spring is converted into the extra kinetic energy of relative motion.

If you run this backwards in time (i.e. reverse the final velocities of the two particles) you have what is usually described as an inelastic collision: the two particles come together, stick, and continue as one particle. In this case, for example, the extra kinetic energy is dissipated as heat.

You can always analyse this kind of problem in a coordinate system which is travelling at constant velocity, $u$, the so-called centre of mass frame. In this frame, it is more obvious what is going on: they suddenly spring apart in opposite directions (or come together).

Total kinetic energy usually increases when a particle 'explodes' and splits into fragments. The 'explosion' occurs because internal energy is released - eg elastic energy stored in a spring, or compressed gas, or chemical energy from a reaction.

You can see that there must be some increase in kinetic energy by running time backwards, so that two particles converging from the right collide and exit together to the left. This is an inelastic collision (some KE is lost) because the relative velocity of separation is zero after the collision. So with time running forwards it is a 'super-elastic' collision (some KE is created).

Drawing a momentum vector triangle helps to show that your assertion about the increase in kinetic energy is correct.

$\vec v_1$ and $\vec v_2$ are the velocities of the fragments after the particle of mass $2m$ splits up.

Pythagoras gives $v_1^2+v_2^2 = 4u^2$ which can be used to find the total kinetic energy of the fragments $\frac 12 m v^2_1 + \frac 12 m v^2_2 = 2 \left ( \frac 12 m u^2\right)$

The vector diagram is the same if it had been the two particles each of mass $m$ colliding and forming a single particle of mass $2m$.
In that case you probably would have been surprised that there was a decrease in kinetic energy and would have called it an inelastic collision with the kinetic energy being converted into heat, sound and doing work in permanently deforming the colliding particles.

The process you describe could have occurred as a result of the explosion of the particle of mass $2m$ into two equal fragments each of mass $m$.
In this case the increased in kinetic energy could be explained as being the result of the chemical energy of the explosive being converted into kinetic energy of the fragments.