From the velocity graph of a car find the accelerations on each stage, the distance and the average speed — Physics problems | PhysLabo
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Distance and average speed from a v(t) graph

Medium 9 grade Mechanics / Motion
Problem statement

The figure shows the graph of the velocity of a car versus time (velocity in m/s, time in seconds). Find the acceleration of the car on each stage of the motion, the distance travelled and the average speed over the whole motion.

Velocity graph of a car: acceleration to 20 m/s in 10 s, constant speed until 30 s, braking to a stop at 40 s
Given
  • 0–10 s: $v$ grows from 0 to 20 m/s
  • 10–30 s: $v = 20$ m/s
  • 30–40 s: $v$ falls from 20 to 0 m/s
  • $a_1, a_2, a_3 = ?$, $s = ?$, $\langle v \rangle = ?$
Hints
Hint 1 / 2
Acceleration is the slope of the $v(t)$ graph: on each straight segment $a = \Delta v / \Delta t$.
Hint 2 / 2
The distance equals the area under the graph: split the figure into two triangles and a rectangle. The average speed is the total distance divided by the total time (not the arithmetic mean of the speeds).
Solution

The acceleration on a stage is the slope of the graph, $a = \Delta v / \Delta t$, and the distance travelled is numerically equal to the area under the velocity graph.

Distance is the area under the graph: S = S₁ + S₂ + S₃
Distance is the area under the graph: S = S₁ + S₂ + S₃

1. Accelerations. On the stage 0–10 s:

$$a_1 = \frac{20 - 0}{10} = 2 \text{ m/s}^2$$

on the stage 10–30 s the velocity is constant, $a_2 = 0$; on the stage 30–40 s (braking):

$$a_3 = \frac{0 - 20}{10} = -2 \text{ m/s}^2$$

2. The distance is the sum of the areas $S_1$ (triangle), $S_2$ (rectangle) and $S_3$ (triangle):

$$S_1 = \frac{1}{2} \cdot 10 \cdot 20 = 100 \text{ m}, \quad S_2 = 20 \cdot 20 = 400 \text{ m}, \quad S_3 = \frac{1}{2} \cdot 10 \cdot 20 = 100 \text{ m}$$$$s = S_1 + S_2 + S_3 = 600 \text{ m}$$

3. The average speed is the total distance divided by the total time:

$$\langle v \rangle = \frac{s}{t} = \frac{600}{40} = 15 \text{ m/s}$$
Answer

$a_1 = 2$ m/s², $a_2 = 0$, $a_3 = -2$ m/s²; $s = 600$ m; $\langle v \rangle = 15$ m/s

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