Relative velocity is a fundamental concept in physics that describes the motion of objects with respect to each other. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach between the two objects. In the context of collisions, relative velocity is an important quantity to consider, especially in elastic collisions where both momentum and kinetic energy are conserved.
Understanding Relative Velocity in Elastic Collisions
In an elastic collision, the relative velocity has certain properties that hold true before and after the collision for any combination of masses. Specifically, the magnitude of the relative velocity is the same before and after the collision, while the relative velocity has opposite signs before and after the collision. This means that if we are sitting on one object moving at a certain velocity, the other object will appear to change direction after the collision, but its speed will remain the same as seen from the first object’s reference frame.
Relative Velocity Theorem
The Relative Velocity Theorem states that in an elastic collision, the relative velocity before the collision is equal in magnitude but opposite in direction to the relative velocity after the collision. Mathematically, this can be expressed as:
$\vec{v}{r,\text{before}} = -\vec{v}{r,\text{after}}$
where $\vec{v}{r,\text{before}}$ is the relative velocity before the collision and $\vec{v}{r,\text{after}}$ is the relative velocity after the collision.
Example: Elastic Collision Between Two Balls
Consider an elastic collision between two balls with masses $m_1$ and $m_2$, moving with initial velocities $\vec{v}_1$ and $\vec{v}_2$, respectively. The relative velocity before the collision is:
$\vec{v}_{r,\text{before}} = \vec{v}_1 – \vec{v}_2$
After the collision, the velocities of the two balls change to $\vec{v}_1’$ and $\vec{v}_2’$, respectively. The relative velocity after the collision is:
$\vec{v}_{r,\text{after}} = \vec{v}_1′ – \vec{v}_2’$
According to the Relative Velocity Theorem, the magnitude of the relative velocity is the same before and after the collision, but the direction is reversed:
$\vec{v}{r,\text{before}} = -\vec{v}{r,\text{after}}$
This means that if the relative velocity before the collision was directed towards the second ball, after the collision, it will be directed away from the second ball.
Calculating Mean Relative Velocity in Gas Collisions
When calculating the mean relative velocity between gas molecules in a collision mean-free path problem, the formula used is:
$\langle |v_r|\rangle =\sqrt2\langle |v|\rangle$
This formula is derived using the Maxwell-Boltzmann distribution of velocities, which describes the distribution of velocities for a large number of particles in a gas. The distribution is given by:
$B(v^2) = 4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2}v^2e^{-mv^2/2k_BT}$
where $m$ is the mass of the gas molecule, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature of the gas.
Since the molecules in the gas are independent, the distribution describing them is the product of the two independent distributions $B(v^2, v’^2)$. This leads to the formula for the mean relative velocity between gas molecules.
Example: Calculating Mean Relative Velocity in Argon Gas
Consider a sample of argon gas at a temperature of 300 K. The mass of an argon atom is $6.63 \times 10^{-26}$ kg. Using the formula for the mean relative velocity, we can calculate the value:
$\langle |v_r|\rangle =\sqrt2\langle |v|\rangle$
where $\langle |v|\rangle$ is the mean speed of the argon atoms, which can be calculated using the Maxwell-Boltzmann distribution:
$\langle |v|\rangle = \sqrt{\frac{8k_BT}{\pi m}}$
Plugging in the values, we get:
$\langle |v|\rangle = \sqrt{\frac{8 \times 1.38 \times 10^{-23} \text{ J/K} \times 300 \text{ K}}{\pi \times 6.63 \times 10^{-26} \text{ kg}}} = 402 \text{ m/s}$
Substituting this into the formula for the mean relative velocity, we get:
$\langle |v_r|\rangle =\sqrt2 \times 402 \text{ m/s} = 568 \text{ m/s}$
So, the mean relative velocity between argon gas molecules at 300 K is approximately 568 m/s.
Relative Velocity in Inelastic Collisions
While the Relative Velocity Theorem holds true for elastic collisions, the relationship between the relative velocities before and after the collision is different for inelastic collisions. In an inelastic collision, the relative velocity after the collision is not simply the negative of the relative velocity before the collision.
In an inelastic collision, the relative velocity after the collision is given by:
$\vec{v}_{r,\text{after}} = \frac{m_1\vec{v}_1′ + m_2\vec{v}_2′}{m_1 + m_2} – \frac{m_1\vec{v}_1 + m_2\vec{v}_2}{m_1 + m_2}$
where $\vec{v}_1’$ and $\vec{v}_2’$ are the final velocities of the two objects after the collision, and $\vec{v}_1$ and $\vec{v}_2$ are the initial velocities before the collision.
This formula takes into account the fact that in an inelastic collision, the final velocities of the two objects are not simply the negative of their initial velocities, as in the case of an elastic collision.
Conclusion
Relative velocity is a fundamental concept in physics that describes the motion of objects with respect to each other. In the context of collisions, relative velocity is an important quantity to consider, especially in elastic collisions where both momentum and kinetic energy are conserved.
The Relative Velocity Theorem states that in an elastic collision, the relative velocity before the collision is equal in magnitude but opposite in direction to the relative velocity after the collision. This property holds true for any combination of masses.
When calculating the mean relative velocity between gas molecules in a collision mean-free path problem, the formula used is $\langle |v_r|\rangle =\sqrt2\langle |v|\rangle$, which is derived using the Maxwell-Boltzmann distribution of velocities.
While the Relative Velocity Theorem holds true for elastic collisions, the relationship between the relative velocities before and after the collision is different for inelastic collisions, where the final velocities of the two objects are not simply the negative of their initial velocities.
Understanding the concept of relative velocity and its properties in different types of collisions is crucial for analyzing and predicting the behavior of objects in various physical systems.
Reference Links:
- How to work out the relation between the “mean relative speed” and the “mean speed” in a gas?
- Relative Velocity – FasterCapital
- Elastic Collisions: Bouncing Back with Momentum
Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess.