Answer:
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
