Question

The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity of this sports car. a) Write the differential equation governing the velocity. Use k for the proportionalty constant (assume k>0 ). b) Solve the differential equation for v(t) such that v(0)=0

Answer 1

Answer:

the equation for v is  v = 250 km/h * [1- e^(-kt) ]

Step-by-step explanation:

since the acceleration "a" is the rate of change of velocity "v" with respect to the time "t", we have

a= dv/dt= k*(250 - v)

thus

dv / (250 - v) = k*dt

if we proceed then to the indefinite integration in both sides

∫dv / (250 - v) = ∫k*dt

- ln (250-v) = k*t +C   , where C= constant

if we know that at t=0 , v(t=0)=0 , then

- ln (250-0) = k*0 +C

- ln 250 = C

replacing C in the original equation:

- ln (250-v) = k*t - ln 250

ln [(250-v)/250] = -k*t

v = 250 km/h * [1- e^(-kt) ]