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ENGR5001 / Week 3
ENGR201 2017 B3Past paperOld spec ENGR2012:125 marks30 min

ENGR201 Engineering Analysis 2017 B3[VALID]

It is the year 1971. A steel container containing nuclear waste, of combined weight WW (N), is dumped into the ocean at time t=0t = 0. Let v(t)v(t) (m/s) be the velocity with which the container moves towards the bottom of the ocean at time tt (s). Let BB be the buoyancy force of the water and DfD_f the drag force, both in newtons and both acting against the motion of the container. If vv is not too large, the drag force is proportional to the speed:

Df=k v(t)(4)D_f = k\,v(t) \qquad (4)

where kk is a constant.

Formulas you may need
  • Newton's second law mdvdt=∑Fm\dfrac{dv}{dt} = \sum F, weight W=mgW = mg (learn this)
  • Laplace of a derivative L[dvdt]=sV(s)−v(0)\mathcal{L}\left[\dfrac{dv}{dt}\right] = sV(s) - v(0) (learn this; printed in the ENGR201 Laplace table)
  • Pairs: 1↔1s1 \leftrightarrow \dfrac{1}{s}, e−αt↔1s+αe^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha} (learn this; printed in the ENGR201 Laplace table)
  • Partial fractions 1s(s+a)=1a(1s−1s+a)\dfrac{1}{s(s + a)} = \dfrac{1}{a}\left(\dfrac{1}{s} - \dfrac{1}{s + a}\right) (learn this)
  • Final Value Theorem lim⁡t→∞f(t)=lim⁡s→0sF(s)\lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s) (learn this)
  1. (a)
    Show that v(t)v(t) is the solution of the differential equation mdv(t)dt+k v(t)=W−B,v(t=0)=0(5)m\frac{dv(t)}{dt} + k\,v(t) = W - B, \qquad v(t = 0) = 0 \qquad (5) where mm is the combined mass of the container (kg). Explain the origin of each term in your answer.
    [5]Third
  2. (b)
    Use the method of Laplace transforms to solve equation (5) for v(t)v(t). From this solution, derive an equation for the steady state velocity of the container.
    [8]2:2
  3. (c)
    Let y(t)y(t) be the depth below the ocean surface attained by the container at time tt. Find y(t)y(t) by integrating v(t)v(t) and using the condition y(0)=0y(0) = 0.
    [5]2:2
  4. (d)
    The container will break if its velocity exceeds 12 m/s when it hits the sea bed. Given that W=3254W = 3254 N, B=3090B = 3090 N and k=0.637k = 0.637 kg/s, show that the container will break if the depth of the ocean where it is dumped exceeds 153 m.
    [7]