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?

Question

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 $a=\frac{v_f-v_i}{t}$.