Question

John walks to the pizza place for lunch. He walks 2 km north then 1 km east. To get home he walks 2 km south and 1 km west. What distance did he travel?
What is his displacement?
How is velocity calculated?
How is acceleration measured?

Answer 1

To analyze John's walk, we'll break down the total distance traveled, his displacement, and then discuss how velocity and acceleration are calculated.

### Total Distance Traveled
- John walks **2 km north** and then **1 km east**.
- To return home, he walks **2 km south** and **1 km west**.

Calculating the total distance:
- **To the pizza place**: \( 2 \, \text{km (north)} + 1 \, \text{km (east)} = 3 \, \text{km} \)
- **Returning home**: \( 2 \, \text{km (south)} + 1 \, \text{km (west)} = 3 \, \text{km} \)

Adding both trips together:
\[
\text{Total distance} = 3 \, \text{km} + 3 \, \text{km} = 6 \, \text{km}
\]

### Displacement
Displacement is defined as the shortest straight-line distance from the starting point to the endpoint, along with the direction.

1. Starting Point: \( (0, 0) \) (origin)
2. Pizza Place: \( (1, 2) \) after walking 2 km north and 1 km east.
3. Ending Point (home): \( (0, 0) \).

The displacement can be calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (0, 0) \)

So,
\[
d = \sqrt{(0 - 0)^2 + (0 - 0)^2} = \sqrt{0} = 0
\]

Therefore, John's displacement ends up being **0 km** because he returns to his starting point.

### How Velocity is Calculated
Velocity is defined as the rate of change of displacement with respect to time. It has both magnitude and direction, making it a vector quantity.

The formula for average velocity \( v \) is:
\[
v = \frac{\text{displacement}}{\text{time}}
\]

If John took \( t \) hours to complete the trip, his velocity would be:
\[
v = \frac{0 \, \text{km}}{t} = 0 \, \text{km/h}
\]

### How Acceleration is Measured
Acceleration is defined as the rate of change of velocity with respect to time. It can also be a vector quantity.

The formula for average acceleration \( a \) is:
\[
a = \frac{\text{change in velocity}}{\text{time interval}}
\]

If John's initial velocity is \( v_i \) and final velocity is \( v_f \), then:
\[
a = \frac{v_f - v_i}{t}
\]

In summary:
- **Total Distance Traveled**: 6 km
- **Displacement**: 0 km (back to starting point)
- **Velocity**: \( 0 \, \text{km/h} \) (over the time taken)
- **Acceleration**: Change in velocity over time, calculated from \( a = \frac{v_f - v_i}{t} \).