UNIVERSITY OF BRITISH COLUMBIA

Science 1 Math/Phys

First Midterm

13 October 1999

Time: 50 minutes

Instructors: Mark MacLean, Jess H. Brewer & Domingo Louis-Martinez

1.
Mathematics   [30 marks]
(a)
Evaluate   ${ \displaystyle \lim_{x\rightarrow 2} \; {x^2+x-6 \over x^2-5x+6} }$ .
(b)
If  f  is differentiable and   ${\displaystyle \lim_{x\rightarrow 1} \frac{f(x)-3}{x-1} = 4 \, }$,   find  f(1)  and  f'(1).
(c)
Compute  h'(0)  if   $h(x)=x^2g(\sin(x))$,  g(0)=1  and  g'(0)=2.

2.
Viscous Drag   [40 marks] When an object moves through a fluid such as air, the flow of the fluid over its surface may be either laminar or turbulent. In laminar flow, the resulting friction of the fluid on the object produces what is known as viscous drag, in which the frictional force  Ff  of the fluid on the object is proportional to the object's velocity  v  through the fluid:   $F_f = - \kappa \, v$,  where  $\kappa$  is a constant. (The  -  sign indicates that the friction force opposes the object's motion, as usual.)

A 100 kg cart is moving in the x direction along a perfectly horizontal road. There is no friction between the cart and the road, but the air exerts a viscous drag force on the cart with force constant   $\kappa = 10$ N-s/m. The cart passes the point  x=0  at  t=0  with a velocity (in the x direction) of  v0 = 10 m/s.

(a)
Plot the acceleration of the cart vs. the velocity of the cart.
(b)
What is the velocity of the cart as a function of time?
(c)
Plot the x position of the cart vs. time.

3.
Molecular Isomers   [30 marks] The cis isomer of a certain molecule has an intrinsic energy $4.1421 \times 10^{-21}$ J higher than the trans isomer of the same molecule. If a large number of these molecules is in thermal equilibrium at a temperature of 300 K, what fraction will be found in the cis state?
[You may need Boltzmann's constant, $k_{\scriptscriptstyle \rm B} = 1.3807 \times 10^{-23}$ J/K.]