Question

The position-time (x-t) graph for a particle moving along x-axis is as shown. Average speed of the particle between \[t = 0\] and \[t = 8\,{\text{s}}\] is

A. Zero
B. 8 m/s
C. 3.75 m/s
D. 4.25 m/s

Answer 1

Hint:The average speed of the particle is the ratio of total distance travelled to the total time taken. The total distance travelled by the particle is the area under the curve of the above figure. The area under the curve consists of two triangles as shown in the above figure.

Formula used:
Average speed, \[{v_{avg}} = \dfrac{D}{t}\]
Here, D is the total distance and t is the total elapsed time.

Complete step by step answer:
We know that the average speed of the particle is the ratio of total distance travelled to the total time taken.
\[{v_{avg}} = \dfrac{D}{t}\]
Here, D is the total distance and t is the total elapsed time.
In the above figure, the total distance travelled by the particle can be calculated by determining the area under the curve. We can see the area under the curve has two triangles of the same base length and different height.
The area under the curve is,
\[D = A = \dfrac{1}{2}\left( {4 \times 5} \right) + \dfrac{1}{2}\left( {4 \times 10} \right)\]
\[ \Rightarrow D = 10 + 20\]
\[ \Rightarrow D = 30\,{\text{m}}\]
Now, the average speed of the particle is,
\[{v_{avg}} = \dfrac{D}{t}\]
Substituting \[D = 30\,{\text{m}}\] and \[t = 8\,{\text{s}}\] in the above equation, we get,
\[{v_{avg}} = \dfrac{{30}}{8}\]
\[ \therefore {v_{avg}} = 3.75\,{\text{m/s}}\]
Therefore, the average speed of the particle between \[t = 0\] and \[t = 8\,{\text{s}}\] is 3.75 m/s.

So, the correct answer is option C.

Note: If you are asked to determine the average velocity of the particle from the given distance-time graph, then the average velocity is zero because the initial and final position of the particle is the same. The average velocity deals with displacement of the particle while the average speed deals with total distance travelled by the particle. Note that the area under the curve is not the average speed of the particle.