 # A car starts moving along a line,first with accelaration a=2m/s^2,starting from rest then uniformally moving and finally deaccelarating at the samerate and comes to rest.The total time of motion is10 seconds.The average speed during the time is 3.2 m/s.How long does the car move uniformly?

5 years ago

The product average velocity and time is the distance travelled by the car = 3.2m/s *10s = 32 m.

The initial velocity of the car = u = 0, uniform acceleration a = 2m/s^2. Then the final velocity = u+at = 0+2t = t.The deceleration being 2m/s^2, the time required to attain the zero velocity from  the initial velocity of  2t is t. Threfore the time of uniform running of the car = 10-2t seconds.

The distance travelled by the car from rest  or the initial velocity of 0 m/s  to reach the final velocity  of 2t with a uniform acceleration of 2m/s^2 = (initial speed+final speed)/2 * time = 0+2t)t= t^2.

The car travels now with uniform speed of 2t m/s for (10-2t) secs. So the distance travelled = 2t(10-2t = 20t-4t^2.

The car now decelerates uniformly at 2m/s^2 and from the initial speed of 2tm/2 till its speed is zero, for this tha car takes t time. Therfore the distance travelled by the car in this time = (initial velocity+final velocity)/2 * time = {(2t+0)/2}*t = t^2.

Thus the distance travelled by the car = t^2+20t-4t^2 +t^2 which should be 32 meter.

Therefore 20t-2t^2 = 32. Or 2t^2-20t+32= 0. Or dividing this equation by 2 , we get: t^-10t +16 = 0. Or (t-8)(t-2) = 0 Therefore t= 2 is the  possible time.

The time during  which the car travels with urniform speed is 10-2t = 10-2*2 = 6 seconds. The uniform speed of the car = 2t = 2*2 = 4m/s. Distance travelled with uniform speed = 4*6 = 32 meter.