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 ).

a) The differential equation for the velocity is given by

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

b) v(t) = 250 - e⁽⁵•⁵² ⁻ ᵏᵗ⁾

With units of km/h

Explanation:

Acceleration, a ∝ (250 - v)

But acceleration is widely given as dv/dt

(dv/dt) ∝ (250 - v)

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

where k = constant of proportionality

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

b) (dv/dt) = k(250 - v)

dv/(250 - v) = k dt

∫ dv/(250 - v) = ∫ k dt

- In (250 - v) = kt + c (where c is the constant of integration)

v(0) = 0; meaning, at t = 0, v = 0

- In 250 = 0 + C

c = - In 250 = - 5.52

- In (250 - v) = kt - 5.52

In (250 - v) = 5.52 - kt

250 - v = e⁽⁵•⁵² ⁻ ᵏᵗ⁾

v = 250 - e⁽⁵•⁵² ⁻ ᵏᵗ⁾

With units of km/h

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