Question

A toy train takes 3 minutes to run one and a half times around a circular track with a circumference of 5.7 m. The difference between its average speed and the magnitude of its average velocity for the one and a half laps is

Answer 1

Solution:

Distance of the circular track= circumference = 5.7 m

Distance covered by the toy train in 3 minutes = one and half circular track * circumference

= 1 1/2 * 5.7m

= 3/2 * 5.7 m

= 8.55m

Time taken to cover one and a half circular track = 3 minutes=180 sec

Now to find average speed and average velocity for one and a half laps .

One laps means one full circle of the track.An example of a laps is one run around a track.

Here one and a half laps means one and a half times around a circular track=8.55m

Formula to find average speed

Average speed = Total distance/total time taken

Average speed to cover one and a half laps = total distance/ total time

= 8.55 m/180 sec

   =0.0475 m/s

Now to find average velocity

Average velocity = Displacement/ total time taken

Here displacement means it is the straight line distance from the initial position to the final position.Its a vector quantity(has both magnitude and direction).

Since the toy train starts at point A and after completing one and a half laps the toy train reaches point B ,which is the exact opposite point of A .Thus the displacement of the toy train after covering one and a half laps will be the diameter of the circle( as initial point is A and final point id B).

  

Thus the dispalcement is diameter of the circle or twice the radius.

Now inorder to find displacement we need to find radius.For that we can use formula

circumference = 2*π * R where R = radius

= 2* 3.14* R π = 3.14

We know circumference = 5.7m.On substituting we get;

5.7 = 2* 3.14* R

R= 5.7/(2* 3.14) =5.7/6.28

R= 0.9076 m

Thus dispalcement = diameter= 2* radius = 2* 0.9076 m = 1.81528 m

Average velocity = Displacement /total time taken to complete one and half laps

= 1.81528 /180 sec

Average velocity= 0.01008 m/sec

NOW we got average speed =0.0475 m/s and Average velocity =0.01008 m/sec

The difference between average speed and average velocity = 0.0475 - 0.01008

= 0.03742 m/s

Thus we can conclude that when a toy train is moving in a circular track of radius 0.9076 m its average speed will be more than average velocity by a value of 0.03742m/s.

NOTE: Average velocity will never be greater than average speed in any situation.It can be equal in some situation but never greater than average speed.