Question

a body moving in a straight line strating from rest accelerate with 2 m/s2 and then retardas with 3 m/s2 finally comes to rest if the total distance for the journey is 30m the total time taken for journey

Answer 1

Given that the body starts to move from rest and moves with an acceleration of 2 ms^(-2), and then retards after sometime with an acceleration of -3 ms^(-2), coming to rest. [-3 is taken because retardation occurs at that time.]

First I've had an assumption due to the fact that the LCM of 2 and 3 is 6.

Since the LCM is 6, I could assume that the body moved with an acceleration of 2 ms^(-2) from rest attaining a velocity of 6 ms^(-1) after a time of 3 seconds, then retarded by an acceleration of -3 ms^(-2) coming to rest after 2 more seconds.

In this case the total time taken is 5 seconds. But what about the time?

Let me draw a velocity - time graph of this body.

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\put(0,-30){\vector(1,0){70}}\put(0,-30){\vector(0,1){70}}\put(0,-30){\line(1,2){30}}\put(30,30){\line(1,-3){20}}\put(72,-35){ X }\put(-5,42){ Y }\put(-5,-35){ O }\multiput(30,30)(0,-5.3){12}{\line(0,-1){2}}\put(30,32){ A }\put(30,-35){ B }\put(50,-35){ C }\end{picture}$

Here the time in s is along x - axis and velocity in ms^(-1) is along the y - axis. The area between the graph and the x - axis gives the displacement of the body in m, i.e., the area of the triangle OAC.

We've assumed the body retarded when it attained a velocity of 6 ms^(-1) for a time 3 seconds. So OB = 3 s and AB = 6 ms^(-1).

Here OC = 5 s.

Displacement = AB × OC / 2 = 6 × 5 / 2 = 15 m

Now we apply something like congruency.

When the body started from rest with an acceleration of 2 ms^(-2) and then retarded with an acceleration of -3 ms^(-2) coming to rest, it could travel a displacement of 15 m in 5 seconds.

If it took 5 seconds to travel 15 m, it could travel 30 m in 10 seconds.

In the previous case the body attains a velocity of 6 ms^(-1) for the first 3 seconds and then comes to rest for 2 more seconds. In this case the body attains a velocity of 12 ms^(-1) for the first 6 seconds and then comes to rest for 4 more seconds.

Hence 10 seconds is the answer.