A car is travelling at 100 km/h on a level road when it runs out of fuel. Its speed v (in km/h) starts to decrease according to the formula d v d t = − k v ( 1 )where k is constant. One kilometre after running out of fuel its speed has fallen to 50 km/h. Use the chain rule substitution d v d t = d v d s d s d t = d v d s vto solve the differential equation.Note: Although I haven't solved it yet, the answers say that this isn't a reasonable model as the velocity is always positive; I didn't make a typo in the question.What I'm trying to do is solve velocity as a function of displacement (s, in km), velocity as a function of time (t, in hours), and displacement as a function of time (I need these functions for later parts of the question).So far I've found velocity as a function of displacement (v(s)): d v d t = − k d s d t (from (1)) ∫ d v d t d t = − k ∫ d s d t d t v ( s ) = − k s + C v ( 0 ) = 100 ⟹ C = 100 , v ( 1 ) = 50 ⟹ k = 50 v ( s ) = − 50 s + 100Then I've tried to find velocity as a function of time (v(t)), but I've got stuck. I can't find any differential equation I can use to get this, or to get displacement as a function of time (s(t)).The answer key says v ( t ) = 100 e − 50 t and s ( t ) = 2 ( 1 − e − 50 t )I've solved such questions many times before, but it's been a while so I'm a bit rusty. So, even a hint might be enough for me to realise what to do.

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A car is travelling at 100 km/h on a level road when it runs out of fuel. Its speed v (in km/h) starts to decrease according to the formula            d      v              d      t        =  −  k  v    (  1  )where k is constant. One kilometre after running out of fuel its speed has fallen to 50 km/h. Use the chain rule substitution            d      v              d      t        =            d      v              d      s                  d      s              d      t        =            d      v              d      s        vto solve the differential equation.Note: Although I haven&#039;t solved it yet, the answers say that this isn&#039;t a reasonable model as the velocity is always positive; I didn&#039;t make a typo in the question.What I&#039;m trying to do is solve velocity as a function of displacement (s, in km), velocity as a function of time (t, in hours), and displacement as a function of time (I need these functions for later parts of the question).So far I&#039;ve found velocity as a function of displacement (v(s)):            d      v              d      t        =  −  k            d      s              d      t          (from (1))  ∫            d      v              d      t        d  t  =  −  k  ∫            d      s              d      t        d  t  v  (  s  )  =  −  k  s  +  C  v  (  0  )  =  100    ⟹    C  =  100  ,     v  (  1  )  =  50    ⟹    k  =  50  v  (  s  )  =  −  50  s  +  100Then I&#039;ve tried to find velocity as a function of time (v(t)), but I&#039;ve got stuck. I can&#039;t find any differential equation I can use to get this, or to get displacement as a function of time (s(t)).The answer key says   v  (  t  )  =  100      e          −      50      t       and   s  (  t  )  =  2  (  1  −      e          −      50      t        )I&#039;ve solved such questions many times before, but it&#039;s been a while so I&#039;m a bit rusty. So, even a hint might be enough for me to realise what to do.

poquetahr · Accepted Answer

We can find v(t), taht k=50.            d      v              d      t        =  −  k  t  =  −  50  t  v  (  t  )  =  A      e          −      50      t        ,         for some constant           A        v  (  0  )  =  100    ⟹    A  =  100  v  (  t  )  =  100      e          −      50      t      We can find s(t) using             d      s              d      t       and the chain rule, however I&#039;m not sure if this is the most efficient method. We know from v(s) that             d      v              d      s        =  −  50, so             d      s              d      v        =  −      1    50  . We can also see from v(t) that             d      v              d      t        =  −  5000      e          −      50      t      .            d      s              d      t        =            d      s              d      v                  d      v              d      t        =  −      1    50    ×  −  5000      e          −      50      t        =  100      e          −      50      t        ∫            d      s              d      t        d  t  =  100  ∫      e          −      50      t        d  t  s  (  t  )  =  100  ×  −      1    50        e          −      50      t        +  C  s  (  t  )  =  −  2      e          −      50      t        +  C  s  (  0  )  =  0    ⟹    C  =  2  s  (  t  )  =  −  2      e          −      50      t        +  2  =  2  (  1  −      e          −      50      t        )

A car is travelling at 100 km/h on a level road when it runs out of fuel. Its speed v (in km/h) star

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