The acceleration of a particle moving along a straight line is given by a = −kt2 m/s2 where k is a constant and time t is in seconds. the initial velocity of the particle at t = 0 is v0 = 12 m/s and the particle reverses it direction of motion at t = 6 s. determine the constant k and the displacement of the particle over the same 6-second interval of motion. ans: k = 1/6 m/s4, δs = 54 m

k = 1/18

Explanation:

Data:

a = -k

to = 0s    Vo = 12m/s

t = 6s the particle chage it's moviment, so v = 0 m/s

We know that acceleration is the derivative of velocity related to time:

rearranging...

Then, we must integrate both sides:

V = 0 because the exercise says that the car change it's direction:

k = 1/6

In order to find X - Xo we must integer v*dT = dX

so...

integrating...

X - Xo = 54m

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