A point mass starts moving in a straight line with constant acceleration "a". at a point after the beginning of motion,the acceleration changes sign, without change in magnitude. determine time T from the beginning of the motion in which the point mass returns to the initial point.

1
by penmetsaramaraj

2015-04-30T01:33:29+05:30

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Equations of motion are:

s = u t + 1/2  a t²  ,
u = 0  and at time t1, when the acceleration changes, distance travelled:
s = 1/2 a t1²
v at t = t1 =  u + a t  = a t1

Now the acceleration is changed to -a.  Then the particle continues in the same direction until the velocity becomes zero.  Then the particle changes the direction and starts accelerating and passes over the point of start.

u = a t1      v = 0    acceleration = -a
v = u + a t
=> 0 = a t1 - a t   =>    t = t1    it takes  t1 more time to stop and reverse direction.

The distance traveled/displacement in this time:
s = u t + 1/2 a t²
=> s = a t1 * t1 - 1/2 a  t1² = 1/2 a t1²

The total displacement from the initial point :  1/2 a t1² + 1/2 a t1² = a t1²

now,  acceleration = -a    u = 0       s = - a t1²   in the negative direction
s = u t + 1/2  a  t²
=> - a t1² = 0 - 1/2 a t²
=>   t = √2 t1

The total time T from initial point forward till back to initial point :
T  =  2 t1 + √2 t1 = (2 + √2) t1

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