Linear Momentum
Introduced| English |
|---|
| linear momentum/ˈlɪnɪə məʊˈmentəm/ |
| vector/ˈvektə/ |
| system of particles/ˈsɪstəm ɒv ˈpɑːtɪklz/ |
A slow truck and a fast bullet
- A truck creeping at walking pace is almost impossible to stop.
- A tiny bullet, moving fast, is also almost impossible to stop.
- Very different objects, yet both carry a lot of the same thing.
- That "quantity of motion" is momentum.
Linear momentum
- Linear momentum 动量 is mass times velocity:
- It measures how hard it is to stop a moving object.
- Both a big mass and a high speed make it large.
A $4\ \text{kg}$ cart moves at $5\ \tfrac{\text{m}}{\text{s}}$. What is its momentum (in kg·m/s)?
$p = mv = 4 \times 5 = 20\ \tfrac{\text{kg}\cdot\text{m}}{\text{s}}$.
Select all ways to increase an object's momentum.
$p = mv$ grows with mass or speed; stopping it drops the momentum to zero.
A vector
- Momentum is a vector 矢量 -- it points the same way as the velocity.
- Direction matters: two objects moving opposite ways have opposite-sign momenta.
- Momenta add as vectors, not just as numbers.
Two objects moving toward each other can have a total momentum of zero.
Momentum is a vector -- opposite directions carry opposite signs, which can cancel to zero.
Momentum is a ____ quantity, pointing the same way as the velocity.
$\vec{p} = m\vec{v}$ is a vector -- it has both size and direction.
Total momentum of a system
- For a system of particles 粒子系统, the total momentum is the vector sum of each one's $m\vec{v}$.
- It also equals the total mass times the velocity of the center of mass.
- This total is what conservation laws will hold fixed.
Linear momentum
Momentum is mass times velocity - at a fixed mass, it grows in step with speed.
A $1\ \text{kg}$ ball at $+4\ \tfrac{\text{m}}{\text{s}}$ and a $2\ \text{kg}$ ball at $-1\ \tfrac{\text{m}}{\text{s}}$. Total momentum (in kg·m/s)?
$p_{tot} = 1(4) + 2(-1) = 4 - 2 = 2\ \tfrac{\text{kg}\cdot\text{m}}{\text{s}}$ -- add the vectors with their signs.
Mass and speed together
- A slow truck and a fast bullet can have the same momentum.
- Big mass, small speed can equal small mass, big speed.
- To change momentum, you must change mass or velocity -- which takes a force over time.
Two identical balls move at equal speeds in opposite directions. Which is zero for the pair?
Momentum cancels (vector); kinetic energy does not (scalar, always positive). So total momentum is zero, energy is not.
A $2\ \text{kg}$ ball moves right at $3\ \tfrac{\text{m}}{\text{s}}$; a $3\ \text{kg}$ ball moves left at $2\ \tfrac{\text{m}}{\text{s}}$.
- Right ball: $p = +6\ \tfrac{\text{kg}\cdot\text{m}}{\text{s}}$; left ball: $p = -6\ \tfrac{\text{kg}\cdot\text{m}}{\text{s}}$.
- Total momentum: $6 - 6 = 0$ -- they cancel because momentum is a vector.
Momentum is not the same as kinetic energy. Momentum ($mv$) is a vector and can cancel; kinetic energy ($\tfrac{1}{2}mv^2$) is a scalar and never cancels. Two objects can have zero total momentum yet plenty of kinetic energy.
Linear momentum $\vec{p} = m\vec{v}$ measures the "quantity of motion" -- a vector pointing along the velocity. The total momentum of a system is the vector sum of the parts. Big mass or high speed makes it large, and opposite directions can cancel.