Conservation of Energy and Momentum

Law of Conservation of Energy

In physics, we focus our attention on a system. There are different types of systems, but particularly especially one is an isolated system.

In an isolated system, the total amount of energy is always the same. This is known as the Law of Conservation of Energy.

Therefore, energy is only converted between different forms. A gain in one type of energy must mean an equal energy loss in another type of energy, such that the total is always the same.

The different forms of energy come under one of two categories: kinetic and potential energy. Thus, the total energy of a system or an object is the sum of its kinetic energy and its potential energy.

Kinetic Energy

Kinetic energy is associated with motion.

We have to transfer energy to an object to get it moving. For instance, when you are riding a bike, your leg muscles are doing work to move forward, and you and the bike gain kinetic energy as you move faster.

Fig. 2 - a cyclist does work with their legs to move forward and gain kinetic energy

There are different types of kinetic energy, such as thermal, sound, and electrical energy.

Potential Energy

Potential energy is associated with a system's arrangement.

Other forms of potential energy include chemical energy and elastic potential energy.

Law of Conservation of Momentum

The linear momentum of an object is a quantity that depends on its mass and velocity.

$$\vec p=m\vec v$$

In the above equation, \( m \) is the mass of the object in \( \mathrm{kg} \) and \( \vec v \) is its velocity in \( \mathrm m/\mathrm s \). The units for momentum, \( \mathrm{\vec p}\) are \( \mathrm{kg\,m}/\mathrm s \). Momentum and velocity are both vectors so their direction must be taken into account. The momentum vector of a moving object points in the same direction as its velocity.

Similarly as with energy, there is even a conservation law for momentum!

Consider a system of two objects that collide. This law implies that the total momentum of the two objects before they collide equals the total momentum after the collision since there are no external forces acting on the system.

Conservation of Momentum Conditions

The only condition for the conservation momentum of a system is that no external forces act on it. However, this does not mean that all forces are forbidden. Momentum is still conserved if there are internal forces acting.

Internal forces cannot change the momentum of a system as they are always canceled out by each other.

Conservation of Energy and Momentum Formulae

We can represent the Law of Conservation of Energy with the following equation:

$$E_{\mathrm i}=E_{\mathrm f},$$

where \( E_{\mathrm i} \) is the initial energy of the system and \( E_{\mathrm f} \) is the final energy.

Remember that the total energy is the sum of the kinetic \( KE \) and potential energy \( PE \) of the system. Let's see how to calculate each of them.

$$\begn{aligne}KE &= \frac12mv^2\\ PE_{\text{gravitaitonal}}&=mgh$$

where \( m \) is the mass of the object in \( \mathrm{kg} \) and \( v \) is its speed in \( \mathrm m/\mathrm s \), \( g \) is the acceleration due to Earth's gravity, \(9.8 \mathrm{m/s^2}\), and \( h \) is the height of the object in meters (\( \mathrm{m} \)).

Similarly, the law of conservation of momentum can be represented by the formula:

$$\vec{p_{\mathrm i}}=\vec{p_{\mathrm f}},$$

As discussed above, momentum is a vector quantity so you must pay attention to the direction of the object's motion when calculating its momentum.

Difference Between Conservation of Energy and Conservation of Momentum

Both conservation principles are very similar but they have an important difference. Energy is a scalar quantity; it has a numerical value and no direction. On the other hand, momentum is a vector quantity and so it has a direction. Usually, we use the coordinate system with the \( x \), \( y \) and \( z \) axes. The momentum of an object can be calculated along each axis and it will be conserved in each direction, independently.

Fig. 6 - The velocity vector (and hence the momentum vector) can be split up into components in three perpendicular directions.

Conservation of Energy and Momentum in Collisions

The laws of conservation of energy and momentum are useful when dealing with collisions. While momentum is always conserved in collisions (as long as no external forces exist!) energy can be wasted in some types of collisions.

Elastic Collisions

No energy is lost in elastic collisions. All of the energy is kinetic.

The general equations for the final velocities of particles colliding in an elastic collision can be found from the equations for conservation of momentum and kinetic energy. Consider an elastic collision between particle \( 1 \) with a mass \( m_1 \) and particle \( 2 \) with a mass \( m_2 \). They collide along a straight line - they do not change direction. Particle \( 1 \) has an initial speed of \( u_1 \) and a final speed of \( v_1 \). Particle \( 2 \) has an initial speed of \( u_2 \) and a final speed of \( v_2 \). This means that we can write the conservation of momentum equation as

$$m_1u_1+m_2u_2=m_1v_1+m_2v_2.$$

The collision is elastic so the kinetic energy before and after is the same, so we can also write

$$m_1u_1^2+m_2u_2^2=m_1v_1^2+m_2v_2^2.$$

Inelastic Collisions

Kinetic energy is lost in all inelastic collisions. However, kinetic energy is lost the most at completely inelastic collisions where the objects stick together after they collide.

Consider a completely inelastic collision between two particles, with the same notation as before. However, this time the particles stick together after the collision so they have the same final velocity \( V \). The equation of conservation of momentum for this case becomes

$$m_1u_1+m_2u_2=(m_1+m_2)V.$$

And we can solve for the final velocity of the particles:

$$V=\frac{m_1u_1+m_2u_2}{m_1+m_2}.$$

The Ballistic Pendulum

A ballistic pendulum is a device used for measuring the speed of a projectile using the conservation of energy and momentum. The projectile is fired into a large block of wood suspended by two cords, which causes the block to swing with the motion of a pendulum. There are angle markings that follow the path of the block's swing so that the height of the swing can be calculated.

Fig. 8 - a ballistic pendulum must be carefully set up to measure the correct speed of a bullet

Suppose that a bullet of mass \( m \) reaches the block with a speed \( u \) (which is what we want to find) and the wooden block has a mass \( M \). The bullet comes to a complete stop within the block, so the collision is completely inelastic. We can use the equation derived above for the final speed of two objects after a completely inelastic collision:

$$V=\frac{m_1u_1+m_2u_2}{m_1+m_2}.$$

Let the speed of the bullet and block after the collision be \( v \) so that this equation gives

$$v=\frac{mu}{m+M}$$

After the impact, the block and bullet together have kinetic energy.

$$\begin{aligned}KE&=\frac12mv^2\\[8pt] KE&=\frac12(m+M){\left(\frac{mu}{m+M}\right)}^2\\[8pt]KE&=\frac12\frac{m^2u^2}{m+M}\end{aligned}$$

As they begin to rise, this kinetic energy is converted into gravitational potential energy. Once the block is at the end of its pendulum motion, at its highest point, \( h \), all of the initial kinetic energy will have been converted to gravitational potential energy.

$$PE_{\text{gravitational}}=(m+M)gh,$$

The potential energy can be equated to the initial kinetic energy of the block and bullet to find the bullet's speed!

$$\begin{aligned}KE&=PE\\[8pt]\frac12\frac{m^2u^2}{m+M} &=(m+M)gh\\[8pt]u^2&=\frac{2{(m+M)}^2gh} {m^2}\\[8pt] u&=\frac{(m+M)}m\sqrt{2gh}\end{aligned}$$