Antiderivatives: Car Deceleration ProblemA car braked with a constant deceleration of 40 ft / s 2 , producing skid marks measuring 160 ft before coming to a stop. How fast was the car travelling when the brakes were first applied?I know I can solve this problem using kinematics equations from physics; using v f 2 = v i 2 + 2 a d yields an initial velocity of 113 ft/s. However, I am supposed to be using antiderivatives and not physics. So far, I figured that if a ( t ) = − 40 then v ( t ) = − 40 t + c 1 and d ( t ) = − 20 t 2 + c 1 x + c 2 , where c 1 and c 2 are constants. I'm not quite sure what my next step should be... any suggestions?

Question

Antiderivatives: Car Deceleration ProblemA car braked with a constant deceleration of   40   ft      /        s    2  , producing skid marks measuring 160 ft before coming to a stop. How fast was the car travelling when the brakes were first applied?I know I can solve this problem using kinematics equations from physics; using       v    f    2    =      v    i    2    +  2  a  d yields an initial velocity of  113 ft/s. However, I am supposed to be using antiderivatives and not physics. So far, I figured that if   a  (  t  )  =  −  40 then   v  (  t  )  =  −  40  t  +      c    1   and   d  (  t  )  =  −  20      t    2    +      c    1    x  +      c    2  , where       c    1   and       c    2   are constants. I&#039;m not quite sure what my next step should be... any suggestions?

popman14ee · Accepted Answer

Explanation:Your equations are correct ( with a t instead of x in the second equation) and, if we measure the time and the space from the instant in which began the braking, in your second equation we have       c    2    =  0 ( the initial position is   d  (  0  )  =  0). the other constant       c    1   is the inital velocity   v  (  0  )  =      c    1   and at the instant       t    1   in wich the car stops its motion we have:      {                            0          =          −          40                      t            1                    +          v          (          0          )                                      160          =          −          20                      t            1            2                    +          v          (          0          )                      t            1                                   solve this system and you find the duration of the braking       t    1   and the inital velocity v(0).