A body of constant mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k | v | , where k is a constant. Neglect changes in the gravitational force.
i) Find the maximum height xm attained by the body and the time tm at which this maximum height is reached.
ii) Show that if kv0 /mg < 1, then tm and xm can be expressed as
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The following table gives the sales of an item produced by a company since 1999.
| Year | Sales (in crores of rupees) |
| 1999 2000 2001 2002 2003 2004 | 10 12 9 13 12 14 |
Consider the following first-order ODE formulation = a{L- n(t)}, n(t0) = n0
.
Associate the variables {t, n(t)} and the parameters {a, L} to a physical system so that the above formulation becomes a mathematical model for population changes.
A particle is moving under a central force along the path given by the equation r = a tan .Show that the radial acceleration is
where,
Let a, K and 0 N be positive real numbers such that 0 < N0 <K. Find the solution of the logistic equation
If N0, N1, and N2 be the values of N at t = 0, t = T and t = 2T respectively, then show that
Also if at time t = t , the population N reaches half its carrying capacity then what is the form of the solution of the logistic equation?
State the type of modelling you will use for the given problem, giving reasons for your answers. Also state four essentials for the problem.
A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sales for the past 5 years are available. To obtain good estimates for future demand so as to help the company make the right decisions.
Television sets for repair arrive at random at an average of 4 per day to a single repairman
who takes an average of hours to carry out each repair. It being assumed that the repair times have an exponential distribution. What is the average number of television sets in the workshop? What is the probability that an arriving set will find at least 3 sets in front of it?
The repairman works for eight hours a day.
Suppose we find the sum of solutions obtained in (i) and (ii) of (b) above. Does this give a solution of the equation describing the motion? If yes, what physical situation, this solution will be correspond to? When is the velocity of the pendulum maximum? What type of energy does the pendulum have at position of maximum velocity?