1. Why talk about energy instead of just forces?

• Describing motion with Newton's laws is sometimes hard (lots of forces, directions, time-steps).
• With energy we can treat whole motions in one step: add up the energies before and after.
• Key idea: Energy can change form (kinetic → potential, etc.) but the total in a closed system stays the same.

---

2. Common forms of mechanical energy

Other forms (chemical, nuclear, electrical…) exist but the three above plus thermal cover most mechanics problems.

---

3. Systems, boundaries, and the Energy Principle

• System = the matter you choose to analyze. Draw an imaginary boundary around it.
• Inside the boundary, energies can transform; across the boundary, energy can transfer by work (mechanical) or heat (thermal, beyond this chapter).
• Energy Principle (book's wording)

$$W_{\text{ext}}=\Delta E_{\text{sys}}$$

"Work done on the system by the outside equals the change in the system's total energy."

---

4. Work $W$: the currency of energy transfer

• Formal definition

$$W=\int_{s_i}^{s_f} F_{\parallel}\,ds \;=\;\vec F\cdot\Delta\vec r$$

(for a constant force that is the familiar dot product).

• Only the component of the force parallel to the motion matters.
• Sign:
  • $0°<\theta<90°$ $\implies$ positive work (adds energy).
  • $90°<\theta<180°$ $\implies$ negative work (removes energy).
  • $\theta=90°$ or no displacement $\implies$ zero work.
• Units: joule (J) = N · m = kg · m²/s².

---

5. The Work–Kinetic‑Energy Theorem

For a point mass:

$$W_{\text{tot}} \;=\; \Delta K$$

Total work done on the object equals the change in its kinetic energy. This is just Newton's 2nd law in energy clothing.

---

6. Potential energies in detail

6.1 Gravitational

Choose a convenient reference level (often $y=0$ at ground/table). Then

$$U_G = m g y$$

It can be positive (above the reference) or negative (below).

6.2 Elastic (spring)

• Hooke's law: $\vec F_{Sp} = -k\,\Delta s\,\hat s$ (force points toward equilibrium).
• Stored energy:

$$U_{Sp} = \tfrac12 k\,(\Delta s)^2$$

Zero when the spring is at its relaxed length.

---

7. Dissipative forces & thermal energy

Kinetic friction, air drag, etc. don't store energy—they convert mechanical energy into thermal:

$$\Delta E_{th} = F_{diss}\,\Delta s \quad(\text{always }+)$$

So mechanical energy ($K+U_G+U_{Sp}$) drops by that same amount.

---

8. Power $P$: how fast energy moves

• Instantaneous:

$$P = \frac{dE_{\text{sys}}}{dt} \;=\; \vec F\cdot\vec v$$

• Average: $P_{\text{avg}}=\dfrac{\Delta E}{\Delta t}=\dfrac{W}{\Delta t}$
• Units: watt (W) = J/s. Positive when delivering energy, negative when absorbing it.

---

9. Practice problems

Try each one first; click Reveal Answer to uncover a full solution.

---

Problem 1 Vector work on a catamaran

A wind pushes with $\vec F_1=(-25\hat{\imath}-15\hat{\jmath})$ N. Water resists with $\vec F_2=(13\hat{\imath}-17\hat{\jmath})$ N. During a tack the displacement is $\Delta\vec r=(-260\hat{\imath}+340\hat{\jmath})$ m. Find the total work done on the craft.

---

Problem 2 Rocket on a spring

A 12 kg rocket sits on a vertical spring $k=560$ N/m, initially compressed 0.21 m and at rest. Engine fires → rocket moves upward with 1.8 m/s while the spring is now stretched 0.40 m. Neglect friction. How much chemical energy did the engine supply?

---

Problem 3 Dog-sled power

A 220 kg sled starts from rest. A dog team pulls with a constant force giving $a=0.75$ m/s² until the sled reaches 3.3 m/s; friction is negligible.

• (a) Find the team's average power for the whole run.
• (b) Find their instantaneous power the moment the sled hits 3.3 m/s.

---

10 Key take-aways

• Think energy for "before vs. after" questions, forces for "what happens right now."
• Identify your system, list its energy forms, then apply

$$W_{\text{ext}} = \Delta(K+U_G+U_{Sp}+E_{th}+…)$$

• For constant forces: work is easy dot-product geometry.
• Power tells you "How quickly?"—vital in engineering and physiology.