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# Step-by-Step Solution

Let's break down the problem into clear steps.

## **Given Data**
- Mass ($ m $) = **120 kg**
- Frontal area ($ A $) = **2.5 m²**
- Drag coefficient ($ C_w $) = **.42**
- Air density ($ \rho $) = **1.25 kg/m³**
- Slope = **10%** ($\theta = 5.71^\circ$)
- Speed ($ v $) = **72 km/h** = **20 m/s** (since $ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} $)

---

## **1. Rolling Resistance Force ($ F_{rr} $)**

Use coefficient of rolling resistance ($ C_{rr} $), which is typically about **.015** for cars (assumed since not given).

$$
F_{rr} = C_{rr} \cdot m \cdot g \cdot \cos(\theta)
$$

$$
F_{rr} = .015 \times 120 \times 9.81 \times .995
$$
$$
F_{rr} \approx .015 \times 120 \times 9.81 \times .995 \approx 176 \text{ N}
$$

---

## **2. Grade Resistance Force ($ F_{grade} $)**

$$
F_{grade} = m \cdot g \cdot \sin(\theta)
$$
$$
F_{grade} = 120 \times 9.81 \times \sin(5.71^\circ)
$$
$$
\sin(5.71^\circ) \approx .0995
$$
$$
F_{grade} = 120 \times 9.81 \times .0995 \approx 1172 \text{ N}
$$

---

## **3. Aerodynamic Drag Force ($ F_{aero} $)**

$$
F_{aero} = \frac{1}{2} \cdot \rho \cdot C_w \cdot A \cdot v^2
$$
$$
F_{aero} = .5 \times 1.25 \times .42 \times 2.5 \times (20)^2
$$
$$
= .5 \times 1.25 \times .42 \times 2.5 \times 400
$$
$$
= .625 \times .42 \times 2.5 \times 400
$$
$$
= .2625 \times 2.5 \times 400
$$
$$
= .65625 \times 400
$$
$$
= 262.5 \text{ N}
$$

---

## **4. Total Tractive Force ($ F_{tractive} $)**

$$
F_{tractive} = F_{rr} + F_{grade} + F_{aero}
$$
$$
F_{tractive} = 176 + 1172 + 262.5 = 161.5 \text{ N}
$$

---

## **5. Power Required ($ P $)**

$$
P = F_{tractive} \cdot v
$$
$$
P = 161.5 \times 20 = 32,210 \text{ W} = 32.2 \text{ kW}
$$

---

# **Final Answers**

## **Tractive Force:**  
$$
\boxed{161.5 \text{ N}}
$$

## **Power at the wheels:**  
$$
\boxed{32.2 \text{ kW}}
$$

---

### **Summary Table**

| Quantity            | Value           |
|---------------------|----------------|
| Tractive Force      | 161.5 N       |
| Power at Wheels     | 32.2 kW        |

If you need the effect of gear ratio or more details, let me know!

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