A metal ball with the momentum mv strikes a wall and bounces back. The change in the ball's momentum is ideally
-2mv
Understanding Change in Momentum During Collision
Let's analyze what happens to the momentum of the metal ball when it strikes a wall and bounces back. Momentum is a vector quantity, which means it has both magnitude and direction. It is defined as the product of mass and velocity ($p = mv$). When the ball hits the wall and bounces back, its velocity changes direction, even if its speed remains the same (in an ideal elastic collision).
Initial and Final Momentum Vectors
We are given that the initial momentum of the ball is $mv$. Let's assume the ball is moving towards the wall in the positive direction. So, the initial momentum can be represented as:
$\vec{p}_{initial} = +mv$
When the ball bounces back, its speed is ideally the same as the initial speed, but its direction is reversed. If the initial direction was positive, the final direction will be negative. Therefore, the final momentum is:
$\vec{p}_{final} = -mv$
This change in velocity during the collision with the wall is what causes the change in momentum.
Calculating the Change in Momentum
The change in momentum ($\Delta \vec{p}$) is defined as the final momentum minus the initial momentum:
$\Delta \vec{p} = \vec{p}_{final} - \vec{p}_{initial}$
Substituting the values we determined:
$\Delta \vec{p} = (-mv) - (+mv)$
$\Delta \vec{p} = -mv - mv$
$\Delta \vec{p} = -2mv$
The result, $-2mv$, represents the vector change in momentum. The negative sign indicates that the change in momentum is in the direction opposite to the initial momentum.
| Quantity | Description | Value |
|---|---|---|
| Initial Momentum ($\vec{p}_{initial}$) | Momentum before hitting the wall (assuming positive direction towards wall) | $+mv$ |
| Final Momentum ($\vec{p}_{final}$) | Momentum after bouncing back (assuming negative direction away from wall) | $-mv$ |
| Change in Momentum ($\Delta \vec{p}$) | Final Momentum − Initial Momentum | $-2mv$ |
This calculation shows that the change in the ball's momentum is indeed $-2mv$. This concept is related to impulse, which is equal to the change in momentum.
Therefore, the ideal change in the ball's momentum during this bouncing ball scenario is $-2mv$. This is a classic physics problem illustrating the vector nature of momentum and its change during a collision.
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 $\vec{P}$?
