Problem 1
Traffic on a northbound highway segment is stationary and is composed of two families of vehicles; cars which
travel at speed vc and trucks which travel at speed vt . A moving observer traveling in the opposing (i.e.,
southbound) direction travels a distance L in time T, L / T = v o
. During this trip, the moving observer counts
the number of (northbound) cars, mc, and the number of (northbound) trucks, mt, that pass her.
Derive an expression for the total flow of vehicles in the northbound direction in terms of the variables given.

To solve this problem, we need to derive an expression for the total flow of vehicles in the northbound direction in terms of the given variables.

### Variables:

- \( v_c \): speed of cars (in the northbound direction).
- \( v_t \): speed of trucks (in the northbound direction).
- \( v_o \): speed of the observer (in the southbound direction), \( v_o = \frac{L}{T} \).
- \( m_c \): number of cars observed passing by the observer during the time \( T \).
- \( m_t \): number of trucks observed passing by the observer during the time \( T \).
- \( T \): the time during which the observer counts the vehicles.
- \( L \): the distance the observer travels in time \( T \).

### Understanding the problem:
The observer, traveling in the southbound direction, sees cars and trucks traveling northbound. The observer counts \( m_c \) cars and \( m_t \) trucks over the time \( T \). We are interested in determining the total flow of vehicles (i.e., the number of cars and trucks passing a fixed point per unit time) on the northbound highway.

#### 1. Apparent Relative Speeds:
When the observer sees a northbound car or truck, the relative speed between them and the observer is the sum of their speeds (since they are moving in opposite directions):
- The relative speed between the observer and a car: \( v_c + v_o \).
- The relative speed between the observer and a truck: \( v_t + v_o \).

#### 2. Counting Cars and Trucks:
During the time \( T \), the observer counts \( m_c \) cars and \( m_t \) trucks. The number of cars and trucks passing the observer is related to the density of cars and trucks on the road.

Let:
- \( \lambda_c \): the density of cars (number of cars per unit distance).
- \( \lambda_t \): the density of trucks (number of trucks per unit distance).

The total number of cars and trucks passing the observer is:
\[
m_c = \lambda_c \times (v_c + v_o) \times T
\]
\[
m_t = \lambda_t \times (v_t + v_o) \times T
\]

From these equations, we can solve for the densities \( \lambda_c \) and \( \lambda_t \):
\[
\lambda_c = \frac{m_c}{(v_c + v_o)T}
\]
\[
\lambda_t = \frac{m_t}{(v_t + v_o)T}
\]

#### 3. Flow of Vehicles:
The flow of vehicles (in the northbound direction) is the number of vehicles passing a point per unit time.

- Flow of cars: \( F_c = \lambda_c \times v_c \).
- Flow of trucks: \( F_t = \lambda_t \times v_t \).

Substitute the expressions for \( \lambda_c \) and \( \lambda_t \):
\[
F_c = \left( \frac{m_c}{(v_c + v_o)T} \right) \times v_c = \frac{m_c v_c}{(v_c + v_o) T}
\]
\[
F_t = \left( \frac{m_t}{(v_t + v_o)T} \right) \times v_t = \frac{m_t v_t}{(v_t + v_o) T}
\]

#### 4. Total Flow:
The total flow \( F \) of vehicles in the northbound direction is the sum of the flow of cars and the flow of trucks:
\[
F = F_c + F_t = \frac{m_c v_c}{(v_c + v_o) T} + \frac{m_t v_t}{(v_t + v_o) T}
\]

### Final Expression for Total Flow:
\[
F = \frac{m_c v_c}{(v_c + v_o) T} + \frac{m_t v_t}{(v_t + v_o) T}
\]
This is the expression for the total flow of vehicles in the northbound direction in terms of the given variables.

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Would you like more details on any part of the solution or have any questions?

Here are some follow-up questions to explore this concept further:

1. How would the expression for total flow change if the observer was stationary?
2. How does the flow of vehicles change if the observer's speed \( v_o \) increases?
3. What assumptions are made about the distribution of cars and trucks on the highway?
4. How would this expression change if vehicles could also overtake each other?
5. Can you derive the flow if there was only one type of vehicle on the highway?

**Tip:** Relative velocity plays a crucial role in calculating the number of vehicles passing by an observer moving in the opposite direction.

Traffic on a northbound highway segment is stationary and is composed of two families of vehicles; cars which travel at speed vc and trucks which travel at speed vt. A moving observer traveling in the opposing (i.e., southbound) direction travels a distance L in time T, L / T = v o. During this trip, the moving observer counts the number of (northbound) cars, mc, and the number of (northbound) trucks, mt, that pass her. Derive an expression for the total flow of vehicles in the northbound direction in terms of the variables given.

Explore how to calculate the total flow of cars and trucks on a northbound highway. This advanced problem involves relative velocity concepts, the density of vehicles, and key flow formulas. Perfect for undergraduate students or high schoolers in Grades 11-12.

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