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The problem concerns a car (Pkw) moving through a valley and then over a hill, with a given initial speed and radius of curvature for both sections. Let's analyze each question step-by-step:

### Given Data:
- \( r_1 = 135 \, \text{m} \) (radius of the valley)
- \( r_2 = 68 \, \text{m} \) (radius of the hill)
- \( v_0 = 72 \, \text{km/h} = 20 \, \text{m/s} \) (speed in meters per second)
- \( m = 80 \, \text{kg} \) (mass of the driver)

The key forces here include:
- The **gravitational force** \( F_G \) on the driver, given by \( F_G = m \cdot g \), where \( g = 9.81 \, \text{m/s}^2 \).
- The **centripetal (radial) force** required for circular motion in the valley and over the hill.
- The **normal force** (constraint force) exerted by the seat on the driver, which varies based on the direction of the centripetal force (toward the center of curvature in each scenario).

Let's solve each part:

---

### Part (a) - Weight of the Driver
The weight \( F_G \) of the driver is calculated as:
\[
F_G = m \cdot g = 80 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 784.8 \, \text{N}
\]

---

### Part (b) - Radial and Constraint Forces in the Valley (Talsenke)

In the valley, the centripetal force \( F_{r1} \) required for circular motion is:
\[
F_{r1} = \frac{m \cdot v_0^2}{r_1} = \frac{80 \cdot (20)^2}{135} = \frac{80 \cdot 400}{135} = 237.04 \, \text{N}
\]

The **constraint force \( F_{Z1} \)** in the valley is the sum of the gravitational force and the centripetal force (since both act in the same direction toward the center of curvature):
\[
F_{Z1} = F_G + F_{r1} = 784.8 + 237.04 = 1021.84 \, \text{N}
\]

---

### Part (c) - Radial and Constraint Forces on the Hill (Bergkuppe)

On the hill, the centripetal force \( F_{r2} \) required for circular motion is:
\[
F_{r2} = \frac{m \cdot v_0^2}{r_2} = \frac{80 \cdot (20)^2}{68} = \frac{80 \cdot 400}{68} = 470.59 \, \text{N}
\]

The **constraint force \( F_{Z2} \)** on the hill is the gravitational force minus the centripetal force (since they act in opposite directions):
\[
F_{Z2} = F_G - F_{r2} = 784.8 - 470.59 = 314.21 \, \text{N}
\]

---

### Part (d) - Speed at which the Car Loses Ground Contact on the Hill

For the car to lose contact with the ground, the normal force \( F_{Z2} \) must be zero. This condition implies:
\[
F_G = F_{r2}
\]
Thus:
\[
m \cdot g = \frac{m \cdot v^2}{r_2}
\]
Solving for \( v \):
\[
v = \sqrt{g \cdot r_2} = \sqrt{9.81 \cdot 68} = \sqrt{667.08} \approx 25.83 \, \text{m/s}
\]
Converting to km/h:
\[
v \approx 25.83 \times 3.6 \approx 93 \, \text{km/h}
\]

---

### Summary of Results:
- (a) Weight of the driver, \( F_G = 784.8 \, \text{N} \).
- (b) Radial force in the valley, \( F_{r1} = 237.04 \, \text{N} \); constraint force, \( F_{Z1} = 1021.84 \, \text{N} \).
- (c) Radial force on the hill, \( F_{r2} = 470.59 \, \text{N} \); constraint force, \( F_{Z2} = 314.21 \, \text{N} \).
- (d) Speed at which the car loses ground contact on the hill, \( v \approx 93 \, \text{km/h} \).

Would you like more details or have any further questions?

---

### Related Questions:
1. How would the forces change if the driver's mass increased?
2. What would happen to the normal force if the speed was doubled?
3. How does the radius of curvature affect the radial force in each case?
4. Why does the car lose contact at the hill and not in the valley?
5. What factors could affect the value of \( g \) and how would that impact the results?

### Tip:
When dealing with circular motion, always check the direction of forces involved to determine whether they add up or subtract in scenarios like valleys and hills.

Ein Pkw fährt auf einem kurvenfreien Streckenabschnitt mit der Geschwindigkeit v0 durch eine Talsenke (Krümmungsradius r1) und danach über eine Bergkuppe (Krümmungsradius r2). Der Fahrer hat die Masse m.

(a) Wie groß ist die Gewichtskraft FG des Fahrers?
(b) Wie groß sind Radialkraft Fr1 und Zwangskraft FZ1 für den Fahrer in der Talsenke?
(c) Wie groß sind Radialkraft Fr2 und Zwangskraft FZ2 für den Fahrer auf der Bergkuppe?
(d) Bei welcher Geschwindigkeit v1 verliert der Pkw auf der Bergkuppe die Bodenhaftung?

Gegeben: r1 = 135 m, m = 80 kg, r2 = 68 m, v0 = 72 km/h.

Learn to calculate the weight force of a driver, the radial force, and the constraint force as a car moves through a valley and over a hill at a constant speed. This problem involves circular motion physics concepts, including centripetal and gravitational forces. Suitable for high school students in Grades 11-12.

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