A mass of 4 kg moving at 3 m/s collides with a mass of 6 kg moving at 2 m/s in the opposite direction and they stick together. The combined mass has velocity of:
Zero
Calculating Combined Mass Velocity in an Inelastic Collision
This problem involves a collision where two objects stick together. This type of collision is known as a perfectly inelastic collision. In any collision, whether elastic or inelastic, the principle of conservation of linear momentum always holds true, provided there are no external forces acting on the system.
The law of conservation of linear momentum states that the total linear momentum of an isolated system remains constant over time. In a collision scenario, this means the total momentum of the objects just before the collision is equal to the total momentum of the objects just after the collision.
Let's define the parameters given in the problem:
- Mass of the first object, \(m_1 = 4 \text{ kg}\)
- Velocity of the first object before collision, \(v_1 = 3 \text{ m/s}\)
- Mass of the second object, \(m_2 = 6 \text{ kg}\)
- Velocity of the second object before collision, \(v_2 = 2 \text{ m/s}\) in the opposite direction to \(v_1\).
To apply the conservation of momentum principle, we need to choose a direction as positive. Let's assume the direction of the first mass (\(m_1\)) is positive. Therefore, the velocity of the second mass (\(m_2\)) in the opposite direction will be negative.
- \(v_1 = +3 \text{ m/s}\)
- \(v_2 = -2 \text{ m/s}\)
Since the two masses stick together after the collision, they form a single combined mass. Let \(M\) be the combined mass and \(V\) be their common velocity after the collision.
- Combined mass, \(M = m_1 + m_2\)
- Velocity of the combined mass after collision, \(V\)
The total momentum before the collision is the sum of the individual momenta:
\(p_{\text{before}} = m_1 v_1 + m_2 v_2\)
The total momentum after the collision is the momentum of the combined mass:
\(p_{\text{after}} = (m_1 + m_2) V\)
According to the conservation of linear momentum:
\(p_{\text{before}} = p_{\text{after}}\)
\(m_1 v_1 + m_2 v_2 = (m_1 + m_2) V\)
Now, we can substitute the given values into the equation:
\((4 \text{ kg}) \times (+3 \text{ m/s}) + (6 \text{ kg}) \times (-2 \text{ m/s}) = (4 \text{ kg} + 6 \text{ kg}) \times V\)
Let's perform the calculations:
\(12 \text{ kg} \cdot \text{m/s} - 12 \text{ kg} \cdot \text{m/s} = (10 \text{ kg}) \times V\)
\(0 \text{ kg} \cdot \text{m/s} = 10 \text{ kg} \times V\)
To find the velocity \(V\), we rearrange the equation:
\(V = \frac{0 \text{ kg} \cdot \text{m/s}}{10 \text{ kg}}\)
\(V = 0 \text{ m/s}\)
Thus, the velocity of the combined mass after the inelastic collision is 0 m/s. This means the combined mass comes to rest after they stick together.
Let's verify the given options based on our calculation.
| Option | Velocity |
|---|---|
| 1 | 5.6 m/s |
| 2 | 2.4 m/s |
| 3 | 4.8 m/s |
| 4 | Zero |
Our calculated velocity of 0 m/s matches Option 4.
Revision Table: Key Concepts of Inelastic Collision and Momentum
| Concept | Description | Governing Principle |
|---|---|---|
| Inelastic Collision | A collision where kinetic energy is not conserved. Objects may deform or stick together. | Conservation of Linear Momentum |
| Perfectly Inelastic Collision | An inelastic collision where the colliding objects stick together after the collision, moving as a single combined mass. | Conservation of Linear Momentum (always applies); Kinetic energy is lost. |
| Linear Momentum | The product of mass and velocity (\(\vec{p} = m\vec{v}\)). It is a vector quantity. | Newton's Laws of Motion |
| Conservation of Linear Momentum | In the absence of external forces, the total linear momentum of a system remains constant before, during, and after a collision. | Derived from Newton's Third Law |
Additional Information: Types of Collisions
Collisions in physics are broadly classified based on whether kinetic energy is conserved:
- Elastic Collision: In an elastic collision, both linear momentum and kinetic energy are conserved. Objects rebound after collision without deformation. Examples include collisions between ideal gas particles or a billiard ball hitting another billiard ball (approximately).
- Inelastic Collision: In an inelastic collision, linear momentum is conserved, but kinetic energy is not conserved; some kinetic energy is lost (usually converted into heat, sound, or internal energy). Objects may deform or stick together. Most real-world collisions are inelastic to some degree.
- Perfectly Inelastic Collision: This is a specific type of inelastic collision where the colliding objects stick together and move as a single unit after the collision. This results in the maximum possible loss of kinetic energy while still conserving momentum. The problem we solved is an example of a perfectly inelastic collision.
The principle of conservation of linear momentum is a fundamental concept in physics and is widely applicable in analyzing collisions and other interactions between objects.
Important Questions from Conservation of Linear Momentum
- A particle of mass $6m$ at rest suddenly breaks on its own into three fragments.
Two fragments of mass $m$ and $2m$ move along mutually perpendicular directions with speeds $2v$ and $v$ respectively.
The energy released during the process is, - Body A of mass $m$ moving with speed $u$ collides with another body B of mass $3m$, at rest. The collision is head-on and elastic in nature. After the collision, the fraction of energy lost by the colliding body A is:
What are the dimensions of angular momentum?
The propulsion of a rocket is based on which of Newton's laws of motion?
For a system of interacting particles, which condition is fundamental for the conservation of its total linear momentum $\vec{P}$?
