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A bicyclist has the position-versus-time graph shown in (Figure 1). What is the bicyclist's velocity at  t=35s? Express your answer in meters per second.

To find the bicyclist's velocity at \( t = 35 \, \text{s} \) using the position-time graph provided, we need to calculate the slope of the line segment where \( t = 35 \, \text{s} \) is located, as velocity is the rate of change of position with respect to time.

### Step-by-step:

1. **Identify the relevant section of the graph**:
   - The graph has a steep downward slope from \( t = 30 \, \text{s} \) to \( t = 40 \, \text{s} \).
   - At \( t = 30 \, \text{s} \), the position \( x = 100 \, \text{m} \), and at \( t = 40 \, \text{s} \), the position \( x = 0 \, \text{m} \).

2. **Find the slope**:
   The velocity between \( t = 30 \, \text{s} \) and \( t = 40 \, \text{s} \) is constant because the line is straight. The slope (velocity) is calculated as:
   \[
   v = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1} = \frac{0 \, \text{m} - 100 \, \text{m}}{40 \, \text{s} - 30 \, \text{s}} = \frac{-100}{10} = -10 \, \text{m/s}
   \]

### Answer:
The bicyclist's velocity at \( t = 35 \, \text{s} \) is \( -10 \, \text{m/s} \).

This negative sign indicates that the bicyclist is moving in the opposite direction (back towards the starting point).

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Would you like more details, or do you have any questions? Here are five related questions to deepen understanding:

1. What is the physical interpretation of negative velocity in this context?
2. How would you calculate the total distance traveled by the bicyclist?
3. What is the bicyclist's average velocity from \( t = 0 \, \text{s} \) to \( t = 40 \, \text{s} \)?
4. Can you determine the acceleration at any given point using this graph?
5. How would the graph change if the bicyclist were accelerating instead of moving at a constant velocity?

**Tip**: The slope of a position vs. time graph always gives the velocity at any point in time.

A bicyclist has the position-versus-time graph shown. What is the bicyclist's velocity at t = 35s? Express your answer in meters per second.

This problem explains how to calculate the bicyclist's velocity at t = 35 seconds using a position-versus-time graph. It demonstrates the application of the formula v = Δx / Δt to find the slope of the graph, providing a step-by-step solution suitable for Grades 9-12.

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