Motion and time — distance-time graphs, speed, velocity, acceleration basics

Question

A car travels 120 km in 2 hours, then stops for 30 minutes, then travels 60 km in 1 hour. Draw the distance-time graph and find the average speed for the entire journey.

(CBSE Class 7-9 — Motion and Time)

Motion Type Classification from Graphs

flowchart TD
    A["Distance-Time Graph"] --> B{Shape of graph?}
    B -->|Straight line through origin| C["Uniform speed"]
    B -->|Horizontal line| D["Object at rest"]
    B -->|Curved line (steepening)| E["Increasing speed (acceleration)"]
    B -->|Curved line (flattening)| F["Decreasing speed (deceleration)"]
    C --> G["Slope = speed"]
    D --> H["Speed = 0"]
    E --> I["Slope increasing"]
    F --> J["Slope decreasing"]

Solution — Step by Step

Phase 1: 0 to 2 hours — travels 120 km (speed = 60 km/h)

Phase 2: 2 to 2.5 hours — at rest (0 km, parked for 30 min)

Phase 3: 2.5 to 3.5 hours — travels 60 km (speed = 60 km/h)

The distance-time graph would show:

A straight line with positive slope from (0, 0) to (2, 120)
A horizontal line from (2, 120) to (2.5, 120)
A straight line with positive slope from (2.5, 120) to (3.5, 180)

\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}

Total distance = 120 + 0 + 60 = 180 km

Total time = 2 + 0.5 + 1 = 3.5 hours

\boxed{\text{Average speed} = \frac{180}{3.5} = 51.4 \text{ km/h}}

Note: The average speed (51.4 km/h) is less than the speeds during motion (60 km/h) because the rest period increases the total time without adding distance.

Speed = distance / time (scalar — no direction)

Velocity = displacement / time (vector — has direction)

If the car went 120 km north and then 60 km north, the displacement = 180 km north = distance. But if it went 120 km north and then 60 km south, displacement = 60 km north, while distance = 180 km.

Average speed and average velocity are equal only when the object moves in a straight line in one direction.

Why This Works

A distance-time graph is a visual representation of motion. The slope at any point gives the instantaneous speed. A steeper slope means faster motion. A horizontal section means no motion. Reading graphs is a fundamental skill because it connects the abstract concept of speed to a visual pattern.

Alternative Method — Using the Graph to Estimate

From the graph, we can also find:

Speed during any phase = slope of that segment
Whether the object is speeding up, slowing down, or at rest
Total distance = final y-value minus initial y-value (if no backtracking)

For CBSE Class 9, you also need velocity-time graphs. Key difference: in a d-t graph, slope = speed. In a v-t graph, slope = acceleration, and area under the curve = distance. These two types of graphs are frequently compared in exams — know both.

Common Mistake

Students confuse average speed with the arithmetic mean of speeds. Average speed is NOT $(60 + 0 + 60)/3 = 40$ km/h. It is always total distance divided by total time: $180/3.5 = 51.4$ km/h. The arithmetic mean of speeds only works when the object travels for equal time intervals at each speed — which is rarely the case in real problems.