Question

Consider a head-on elastic collision between two particles. (since the collision is head-on, the motion is confined to a single straight line and is therefore one-dimensional.) Prove that the relative velocity after the collision is equal and opposite to that before. That is, $v_1-v_2=-(v_1^{\prime }-v_2^{\prime }),$ where $v_1$ and $v_2$ are the initial velocities and $v_1^{\prime }$ and $v_2^{\prime }$ the corresponding final velocities.

Short answer

The relative velocity after is equal and opposite to that before: $v_1-v_2=-(v_1^{\prime }-v_2^{\prime }).$

Step-by-step solution

Understand Elastic Collision

In an elastic collision, both momentum and kinetic energy are conserved. We will use these properties to solve the exercise.

Write Momentum Conservation Equation

The total initial momentum is equal to the total final momentum: $m_1{v}_1+m_2{v}_2=m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$.

Write Kinetic Energy Conservation Equation

The total initial kinetic energy equals the total final kinetic energy: $\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2=\frac{1}{2}{m}_1{v}_1^{\prime 2}+\frac{1}{2}{m}_2{v}_2^{\prime 2}$.

Simplify Kinetic Energy Equation

Cancel out the $\frac{1}{2}$ terms: $m_1{v}_1^2+m_2{v}_2^2=m_1{v}_1^{\prime 2}+m_2{v}_2^{\prime 2}$.

Derive Relative Velocity Equation

From the momentum equation, express $v_1^{\prime }$ in terms of $v_2^{\prime }$ by isolating terms: $$v_1-v_1^{\prime }=-(v_2-v_2^{\prime })$$. This simplifies to the relative velocity equation: $$v_1-v_2=-(v_1^{\prime }-v_2^{\prime })$$.

Key concepts

Momentum Conservation

In an elastic collision, momentum conservation is a fundamental principle that ensures the total momentum of the system remains constant before and after the collision. Momentum ($p$) is defined as the product of mass ($m$) and velocity ($v$), and this principle can be mathematically represented for two colliding objects as:

• Initial Total Momentum: $m_1{v}_1+m_2{v}_2$
• Final Total Momentum: $m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$

By conserving momentum, we can set these two expressions equal:$$m_1{v}_1+m_2{v}_2=m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$$This equation helps us understand how the individual velocities of the masses change during the collision while keeping the total momentum constant. It's crucial in predicting the outcome of the collision for both velocities and determining the final state of the system after an elastic collision.
Momentum conservation is foundational in physics because it applies to all systems, regardless of energy changes. This principle allows us to solve complex collision problems by keeping track of linear momentum.

Kinetic Energy Conservation

Kinetic energy conservation is another key aspect of elastic collisions. Unlike inelastic collisions, elastic collisions maintain both the total kinetic energy and momentum. Kinetic energy ($KE$) is related to the motion of an object and is given by the formula:

• Initial Kinetic Energy: $\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2$
• Final Kinetic Energy: $\frac{1}{2}{m}_1{v}_1^{\prime 2}+\frac{1}{2}{m}_2{v}_2^{\prime 2}$

The principle of kinetic energy conservation tells us that in the absence of external forces, the kinetic energies before and after the collision must be equal:$$\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2=\frac{1}{2}{m}_1{v}_1^{\prime 2}+\frac{1}{2}{m}_2{v}_2^{\prime 2}$$Canceling out the $\frac{1}{2}$ terms simplifies our calculations, giving us:$$m_1{v}_1^2+m_2{v}_2^2=m_1{v}_1^{\prime 2}+m_2{v}_2^{\prime 2}$$This equation confirms that the energy used in the collision does not disappear but is instead transferred either within each object or from one object to another.
Understanding kinetic energy conservation enables us to delve deeper into how energy dynamics are maintained and distributed among the colliding bodies during elastic collisions.

Relative Velocity

Relative velocity is a concept that helps us understand the velocity of one object in relation to another. In elastic collisions, a particularly interesting property emerges: the relative velocity of two colliding objects post-collision is equal and opposite to their relative velocity pre-collision. Now, let's break it down:

• Initially, the relative velocity before collision: $v_1-v_2$
• Finally, the relative velocity after collision: $v_1^{\prime }-v_2^{\prime }$

When you hear that these velocities are equal and opposite, it means:$$v_1-v_2=-(v_1^{\prime }-v_2^{\prime })$$This relationship is derived from the conservation laws of both momentum and kinetic energy.
Why is this important? It tells us how the velocities shift in perspective to each other after the collision and reaffirms that energy and momentum are both internally consistent and behave predictably under the conservation laws during elastic collisions.
By grasping this relative change in velocities, students can better predict how objects interact in a collision and provide insight into the effectiveness of conservation laws in predicting outcomes.