III.1.2. Traveled distance. Movement duration

To calculate the distance traveled by a mobile, you need to set a reference point, called the origin.

• If the origin is right in the landmark, we denote by O (that is, x₀ = 0 m).
• If the origin is not exactly in the landmark, we denote by another point, A (for example, x₁ = 20 km from the landmark).

The position of a mobile on a trajectory is the distance from the origin to the mobile, measured on the trajectory. It is denoted by x.

The distance (denoted by d or Δx) traveled by a mobile is the length of the road traveled by the body from a landmark.

d = Δx = x₂ – x₁, dacă x₂ > x₁ (the mobile moves away from the landmark)

d = Δx = x₁ – x₂, dacă x₁ > x₂ (the mobile is approaching the landmark)

🔦 Remark

The distance is always positive, so we subtract the lower position from the higher position.

The duration of the movement (denoted by Δt or t) represents the time interval in which the mobile traveled a certain distance.

Δt = t₂ – t₁

t₁ = the moment of the beginning of the movement and

t₂ = the moment when the movement ends.

🔓 Solved problem

1. A cyclist starts at kilometer 20 at 12:00 and arrives at kilometer 60 at 13:30. How far did the cyclist travel and how long did the movement last?

We write the data of the problem:

Initial position: x₁ = 20 km
Final position: x₂ = 60 km
Initial time: t₁ = 12:00
Final time: t₂ = 13:30

We apply the distance and duration formula and replace the problem data:

d = Δx = x₂ – x₁ = 60 km – 20 km = 40 km

Δt = t₂ – t₁ = 13:30 – 12:00 = 1h 30min = 1,5 h