# Problem-Solving: Mechanics

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/problem-solving-mechanics

## Choosing Your Law

Expert problem-solvers do not start with formulas. They start by choosing a **law**: does this need Newton, energy, or momentum?

| Situation | Best approach |
| --- | --- |
| Find acceleration | **Newton's second law** |
| Find speed at position | **Energy conservation** |
| Collision or explosion | **Momentum conservation** |
| Rotating object | **Torque and $I\alpha$** |
| Oscillation | **SHM equations** |

**General strategy**: Draw a diagram with all forces and axes, list knowns, choose the principle, solve algebraically, then check units.

**Dimensional analysis** requires every equation to be consistent:

$$[F] = \text{kg}\cdot\text{m/s}^2, \quad [E] = \text{kg}\cdot\text{m}^2/\text{s}^2$$

## Pitfalls & Multi-Stage

Mechanics problems often chain different laws across different stages. Watch out for these traps:

- **Vector sums**: Remember $\vec{F}_{\text{net}}$ is a vector sum, not a scalar sum.
- **Non-conservative work**: Do not use energy conservation when non-conservative forces do work.
- **Sign conventions**: Confusing $g = 9.8$ with $g = -9.8$ depends entirely on your axis choice.

> **Collisions**: Applying energy conservation **across** a sticking collision fails. Perfectly inelastic impacts destroy kinetic energy; only momentum bridges that moment. Split the problem at the collision and switch tools.

> **Stuck?** Count equations and unknowns. Fewer equations than unknowns means you need another physical principle.

## Practice questions

8 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Complete the statement with the correct term.

**Answer:** To solve a problem involving forces, the first step is usually to draw a free-**body** diagram.

**Why:** A free-body diagram shows all forces acting on the object of interest.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/complete-the-statement-with-the-correct-term

### 2. Match each problem type to the best conservation law or principle.

**Answer:**

- Collision between two objects → Conservation of momentum
- Object on a roller coaster (no friction) → Conservation of energy
- Object on an incline with friction → Newton's second law
- Satellite in circular orbit → Gravitational force = centripetal force

**Why:** Choosing the right principle depends on what is conserved or what forces act.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/match-each-problem-type-to-the-best-conservation-law-or-principle

### 3. Which strategies are useful for solving mechanics problems?

A. Drawing a free-body diagram
B. Always using energy conservation
C. Checking units in the final answer
D. Choosing a convenient coordinate system

**Answer:** A. Drawing a free-body diagram; C. Checking units in the final answer; D. Choosing a convenient coordinate system

**Why:** Energy conservation is not always applicable (e.g., when non-conservative forces act).

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/which-strategies-are-useful-for-solving-mechanics-problems

### 4. Complete the following passage by choosing the correct terms from the word bank.

**Answer:** On a frictionless incline of angle $\theta$, the component of gravity along the slope is **mg sin θ**, and perpendicular to the slope is **mg cos θ**.

**Why:** Resolving gravity into components along and perpendicular to the incline.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/complete-the-following-passage-by-choosing-the-correct-terms-from-the

### 5. Sort each problem scenario by the most efficient method to solve it.

**Answer:**

- Energy methods: Find speed at bottom of frictionless ramp, Find max height of projectile from speed, Find speed after elastic collision
- Force/Newton's laws: Find normal force on incline, Find acceleration of two connected blocks, Find tension in a string at angle

**Why:** Energy methods skip the details of forces; force methods find individual forces.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/sort-each-problem-scenario-by-the-most-efficient-method-to-solve-it

### 6. A bullet embeds in a hanging wooden block, which then swings upward. To find the bullet’s initial speed, put the solution strategy in the correct order.

**Answer:**

1. Apply momentum conservation to the bullet–block collision
2. Note that kinetic energy is lost while the bullet embeds
3. Apply energy conservation to the upward swing
4. Relate swing height to the speed just after impact
5. Solve backwards for the bullet’s initial speed

**Why:** The classic two-stage analysis: momentum survives the inelastic collision (energy does not), then mechanical energy survives the swing (no more collisions). Choosing the right conservation law for the right stage is the entire art of the ballistic pendulum.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/a-bullet-embeds-in-a-hanging-wooden-block-which-then-swings-upward

### 7. Some mechanics problems are far easier with energy methods than with forces. How can you tell in advance which approach a problem wants?

A. That the forces in the problem are non-conservative
B. That the problem involves more than one object
C. That it asks about start and end states, not the path
D. That the problem gives masses rather than accelerations

**Answer:** C. That it asks about start and end states, not the path

**Why:** Energy relates endpoints; forces describe the journey. A speed at the bottom of a curved track is an energy problem; the time to get there is not.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/some-mechanics-problems-are-far-easier-with-energy-methods-than-with

### 8. When solving mechanics problems, choosing the right coordinate system can simplify the equations significantly.

**Answer:** True

**Why:** Aligning axes with motion or forces reduces the number of vector components.

Page: https://tryals.app/practice/physics-i/problem-solving-mechanics/when-solving-mechanics-problems-choosing-the-right-coordinate-system