Question

a car is traveling along a straight road when the car runs out of gas. the car has an acceleration a(t)=1.1m/s^2-.02m/s^3*t and has a velocity of 30.0m/s at t=0.

a) what is the displacement of the car between t=3.0 s and t=5.0 s?

b) at what time will the car become to rest?

c) what will be the total displacement of the car between t=0 and when it comes to rest?

Answer 1

Let us say that, the car is moving along the x axis,

the acceleration is given by

where

the initial velocity is

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Let us find out the velocity as a funcion of time, the acceleration satisfies

hence

integrating the above expression,

we get the velocity as a function of time

,________________________________________(1)

integrating the above expression we get the position of the car as a function of time,

the position of the car is given by

_________________________________(2)

in the above expression

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Part a

Using (2) we can find the displacement, first let us find the displacement between t=3 and t=0,

now the displacement between t=0 and t=5 is given by (using(2))

therefore the displacement between t=3.0 s and t=5.0 s is

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Part b

The final velocity must be 0, so using (1)

the solution of the above algebraic equation is (the positive one)

t=132.6 s

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part c

The total displacement of the car is given by (using (2))

  

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