1D Elastic Collisions: Momentum and Kinetic Energy Conservation

1. The Two Governing Principles

For any collision in an isolated system, momentum is conserved. For an elastic collision, kinetic energy is also conserved. This gives us the two powerful equations we need.

1. Conservation of Momentum: The total momentum of the system before the collision is equal to the total momentum after.
   $m_1{v}_1+m_2{v}_2=m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$

2. Conservation of Kinetic Energy: The total kinetic energy of the system before the collision is equal to the total kinetic energy after.
   $\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2=\frac{1}{2}{m}_1(v_1^{\prime }{)}^2+\frac{1}{2}{m}_2(v_2^{\prime }{)}^2$

In these equations:

• $m_1$ and $m_2$ are the masses of the two objects.
• $v_1$ and $v_2$ are their initial velocities.
• $v_1^{\prime }$ and $v_2^{\prime }$ are their final velocities (the primes ' indicate "after").

The problem we need to solve is visually represented below. Given the initial conditions, we want to find the final velocities.

Solving this system of two equations directly can be complex because the kinetic energy equation is quadratic (it has squared terms). Fortunately, there's a more elegant way to approach this.

2. Deriving a Simpler Relationship

By combining the two conservation equations through some clever algebra, we can arrive at a much simpler and more intuitive equation. The following video demonstrates this derivation and reveals the final, general formulas for the velocities.

The key insight from that derivation is a beautifully simple relationship between the velocities:
$v_1+v_1^{\prime }=v_2+v_2^{\prime }$
This can be rearranged to:
$v_1-v_2=-(v_1^{\prime }-v_2^{\prime })$

This equation has a clear physical meaning: the relative speed at which the objects approach each other before the collision is equal to the relative speed at which they separate after the collision. The negative sign indicates they are moving apart.

Now we have a new, much easier system of equations to solve:

1. Momentum: $m_1{v}_1+m_2{v}_2=m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$
2. Relative Velocities: $v_1+v_1^{\prime }=v_2+v_2^{\prime }$

This is a system of two linear equations with two unknowns—much more straightforward to solve!

3. Solving a Problem Step-by-Step

Let's put this simplified system to work by solving a typical problem. We will follow a method that you can apply to any one-dimensional elastic collision.

Let's summarize the algorithmic process shown in the video, which you can use as a reliable recipe:

1. Establish a coordinate system: Define a direction as positive (e.g., east or to the right) and the opposite as negative.
2. Write down the knowns: List the masses ($m_1,m_2$) and initial velocities ($v_1,v_2$), being careful with signs.
3. Set up the momentum equation: Substitute the known values into $m_1{v}_1+m_2{v}_2=m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$. This will give you one equation with your two unknowns, $v_1^{\prime }$ and $v_2^{\prime }$.
4. Set up the relative velocity equation: Substitute the known values into $v_1+v_1^{\prime }=v_2+v_2^{\prime }$. This gives you a second, simpler equation with the same two unknowns.
5. Solve the system of two equations. You can use substitution or elimination, just as you would in an algebra class, to find the values of $v_1^{\prime }$ and $v_2^{\prime }$. The signs of your answers will tell you the final direction of motion for each object.

4. Building Intuition with Special Cases

The real power of physics isn't just plugging in numbers, but understanding the patterns that emerge. The general formulas derived in the Physics Ninja video lead to some very interesting results in specific scenarios. Let's look at two common ones.

Case 1: Equal Masses, One Object at Rest

Imagine a billiard ball hitting another identical, stationary ball head-on.

• Setup: $m_1=m_2$, and $v_2=0$.
• Result: The incoming ball stops dead ($v_1^{\prime }=0$), and the target ball moves off with the first ball's original velocity ($v_2^{\prime }=v_1$). They perfectly exchange momentum.

Case 2: A Light Object Hits a Massive Object at Rest

Imagine a golf ball hitting a bowling ball.

• Setup: $m_1\ll m_2$ (mass of object 1 is much less than mass of object 2), and $v_2=0$.
• Result: The light object bounces back with nearly the same speed it came in with ($v_1^{\prime }\approx -v_1$), and the massive object barely moves ($v_2^{\prime }\approx 0$).

These special cases help connect the mathematical formulas back to real-world phenomena you've likely observed.

Conclusion

In this lesson, we've moved from inelastic to elastic collisions. You now have the fundamental tools to predict the outcome when objects collide and bounce apart in one dimension.

Key Takeaways:

• An elastic collision conserves both momentum and kinetic energy.
• This gives us a system of two equations, which we need to solve for the two unknown final velocities.
• The most practical method is to use the conservation of momentum equation along with the simplified relative velocity equation: $v_1-v_2=-(v_1^{\prime }-v_2^{\prime })$.
• This process can be broken down into a repeatable algorithm: set up the two equations, then solve the system algebraically.

Preview of the Next Lesson:

We have now analyzed the two main types of collisions: inelastic and elastic. In our next lesson, we'll take a step back and look at the concept of impulse more closely. We'll explore how momentum and impulse are applied in practical engineering systems designed to manage forces during impacts, such as vehicle airbags and the design of robotic grippers. This will tie everything we've learned about momentum back to your interests in robotics and real-world applications.