Two identical objects move towards each other with velocity 2v and v respectively. What is the velocity of the centre of mass?
Understand the Problem
The question is asking for the velocity of the center of mass of two identical objects moving towards each other with given velocities. To find the center of mass velocity, we will apply the formula for the center of mass in a system of particles and consider the direction of the velocities.
Answer
$v_{cm} = 0$
Answer for screen readers
The velocity of the center of mass is $v_{cm} = 0$.
Steps to Solve
- Identify Object Velocities and Directions
Let's assume the two identical objects have the same mass $m$ and they move towards each other with velocities $v_1$ and $v_2$. For example, let $v_1$ be moving to the right at velocity $+v$ and $v_2$ be moving to the left at velocity $-v$ (the negative sign indicates the opposite direction).
- Set Up the Formula for Center of Mass Velocity
The formula for the velocity of the center of mass ($v_{cm}$) of two objects is given by:
$$ v_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} $$
For identical masses, we denote both masses as $m$. Thus, we can rewrite the formula as:
$$ v_{cm} = \frac{m v_1 + m v_2}{m + m} $$
- Substitute the Values of Velocities
Now we substitute the values of $v_1$ and $v_2$:
$$ v_{cm} = \frac{m (+v) + m (-v)}{m + m} $$
- Simplify the Expression
Since both terms in the numerator share the same mass, we can factor out $m$:
$$ v_{cm} = \frac{m (v - v)}{2m} $$
This simplifies to:
$$ v_{cm} = \frac{0}{2} = 0 $$
- Interpret the Result
The velocity of the center of mass is $0$, indicating that although the two objects are moving towards each other, their center of mass remains at rest because they are identical and moving symmetrically.
The velocity of the center of mass is $v_{cm} = 0$.
More Information
In this case, both objects have equal mass and speed in opposite directions, which causes their center of mass to remain stationary. This scenario is a great illustration of symmetry in physics. It shows that when two identical objects move towards each other with the same speed, their combined motion does not affect the center of mass.
Tips
- Forgetting to consider the direction of the velocities. If one object is moving in the positive direction and the other in the negative direction, it's crucial to assign the correct signs.
- Confusing the center of mass velocity with individual object velocities.
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