The motion of a particle is defined by the relation x 5 t 3 2 9t 2 1 24t 2 8, where x and t are expressed in inches and seconds, respectively. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

a)  t = 2.0 s,  b)  x_f = - 24.56 m,  Δx = 16.56 m

Explanation:

This is an exercise in kinematics, the relationship of position and time is indicated

          x = 5 t³ - 9t² -24 t - 8

a) ask when the velocity is zero

speed is defined by

         v =

let's perform the derivative

        v = 15 t² - 18t - 24

        0 = 15 t² - 18t - 24

let's solve the quadratic equation

       t =

       t1 = -0.8 s

      t2 = 2.0 s

the time has to be positive therefore the correct answer is t = 2.0 s

b) the position and distance traveled for a = 0

acceleration is defined by

       a = dv / dt

       a = 30 t - 18

       a = 0

       30 t = 18

       t = 18/30

       t = 0.6 s

we substitute this time in the expression of the position

       x = 5 0.6³ - 9 0.6² - 24 0.6 - 8

       x = 1.08 - 3.24 - 14.4 - 8

       x = -24.56 m

we see that all the movement is in one dimension so the distance traveled is the change in position between t = 0 and t = 0.6 s

the position for t = 0

       x₀ = -8 m

the position for t = 0.6 s

      x_f = - 24.56 m

the distance

     ΔX = x_f - x₀

     Δx = | -24.56 -(-8) |

     Δx = 16.56 m

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min