# Linear Momentum, Impulse, and Collisions

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/linear-momentum-impulse-and-collisions

## Momentum and Impulse

Airbags save lives by changing the **time** over which momentum drains away. Since $F\Delta t = \Delta p$, ten times the stopping time means one tenth the force.

**Linear momentum** is the product of mass and velocity: $\vec{p} = m\vec{v}$. Newton's second law states $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$.

**Impulse** is the change in momentum produced by a force over time: $\vec{J} = \int \vec{F}\,dt = \Delta\vec{p}$.

**Conservation of momentum**: When no external forces act, total momentum remains constant: $\vec{p}_{\text{total}} = m_1\vec{v}_1 + m_2\vec{v}_2 = \text{constant}$. This holds because internal forces cancel via Newton's third law.

## Collisions and Centre of Mass

Momentum is conserved in **every** collision. What inelastic collisions lose is kinetic energy, never momentum.

| Collision Type | Momentum | Kinetic Energy |
| --- | --- | --- |
| **Elastic** | Conserved | Conserved |
| **Inelastic** | Conserved | Not conserved |
| **Perfectly inelastic** | Conserved | Max loss (objects stick) |

For 1D elastic collisions with equal masses, objects **exchange velocities**.

The **centre of mass** position is $\vec{r}_{\text{cm}} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1+m_2}$. It moves as if all external forces act on a single particle of total mass $M$.

## Practice questions

4 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Which quantities are conserved in an elastic collision?

A. Total mechanical energy
B. Total kinetic energy
C. Total momentum
D. Individual velocities

**Answer:** A. Total mechanical energy; B. Total kinetic energy; C. Total momentum

**Why:** Elastic collisions conserve both momentum and kinetic energy.

Page: https://tryals.app/practice/physics-i/linear-momentum-impulse-and-collisions/which-quantities-are-conserved-in-an-elastic-collision

### 2. Order the stages of a perfectly inelastic collision between two objects.

**Answer:**

1. Two objects approach each other
2. Objects make contact
3. Objects deform and stick together
4. Combined object moves with shared velocity
5. Kinetic energy has decreased (converted to heat/deformation)

**Why:** In perfectly inelastic collisions, objects stick together and KE is lost.

Page: https://tryals.app/practice/physics-i/linear-momentum-impulse-and-collisions/order-the-stages-of-a-perfectly-inelastic-collision-between-two

### 3. An egg thrown at a wall breaks; the same egg thrown just as fast into a sagging bedsheet survives. The egg’s **change in momentum** is the same in both cases. What saves the egg?

A. The sheet absorbs the egg’s momentum before impact
B. The egg slows down before reaching the sheet
C. The sheet stretches the stop over a longer time
D. The sheet reduces the egg’s change in momentum

**Answer:** C. The sheet stretches the stop over a longer time

**Why:** The impulse $F\Delta t$ must equal the fixed $\Delta p$, so stretching the stopping time shrinks the peak force below what cracks the shell. Airbags, crumple zones and bent knees on landing all exploit the same trade.

Page: https://tryals.app/practice/physics-i/linear-momentum-impulse-and-collisions/an-egg-thrown-at-a-wall-breaks-the-same-egg-thrown-just-as-fast-into

### 4. Momentum is conserved in every collision but kinetic energy only in elastic ones. Why do the two conservation laws behave differently?

A. Because momentum is only approximately conserved in inelastic collisions
B. Because internal forces cancel while energy can become heat
C. Because momentum is a vector and kinetic energy is a scalar
D. Because kinetic energy conservation requires the collision to be one-dimensional

**Answer:** B. Because internal forces cancel while energy can become heat

**Why:** Momentum has nowhere else to go; energy does. That is why momentum conservation is the reliable tool for analysing collisions and energy is not.

Page: https://tryals.app/practice/physics-i/linear-momentum-impulse-and-collisions/momentum-is-conserved-in-every-collision-but-kinetic-energy-only-in