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

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) ]