A Random Mechanics Question

(adapted from STEP I, 2009 Q9)

A child plays with a toy cannon on the floor of a long railway carriage, moving north with the acceleration a. The cannon is pointed south at an angle θ to the horizontal. It fires a shell at speed v.

  1. Give an expression for the range of the shell in the carriage. [in terms of v, θ, g and a]
  2. Find the value of tan(2θ) that maximises the range of the shell in the carriage. [hint : its just differentiation]
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Questions for Thermodynamics (2)

HEAT CAPACITIES

  1. With reference to the two different gas manipulations, isochloric and isobaric, explain why two different values of specific molar heat capacity are required to show the relation between the heat flow into a gas and the temperature change it experiences.
    • [hint : in an isochloric expansion, the volume of the gas is fixed and it does no work on its surroundings while being heated. in an isobaric expansion, however, it expands at a constant pressure.]
  2. Let CV be the specific molar heat capacity for an isochloric expansion, and CP be the specific molar heat capacity for an isochloric expansion. Show that CP = CV + R where R is the gas constant.
    • [hint : the work done in an isochloric expansion can be found WD = P dV.]
    • [hint : from the ideal gas equation where P is constant, P dV = nR dT.]
  3. Why does the molar specific heat capacity of a monoatomic gas differ from that of a diatomic gas?
    • [hint :  how do their kinetic energies differ?]

IDEAL GAS MANIPULATIONS 

  1.  On P-V axes for an ideal gas with a constant number of molecules :
    1. Draw an isotherm and define a point (V1, P1) on it.
    2. Draw a line representing the P-V states of the gas as it expands isothermally to V2 > V1.
    3. Shade the area on the graph corresponding to the work done by the gas as it expands isothermally from V1 to V2. Denote this area A.
    4. Give an expression for the amount of heat flow into the gas as it undergoes the previous transition. [hint : change in internal energy of the gas is zero. the WD integral can be evaluated using logarithmic forms.]
    5. Draw a line representing the P-V states of the gas as it expands adiabatically from (V1, P1) to (V2, P3).
    6. Shade the area on the graph corresponding to the work done by the gas as it expands adiabatically from V1 to V2. Denote this area B.
    7. Explain why A > B.
  2. A fixed amount of ideal gas is placed in a one chamber of a bipartite container and thermally isolated from its surroundings. A shutter separates the two, and one of the chambers is evacuated (all molecules are removed from it).  The shutter is then opened and the gas allowed to pass into the other chamber. State :
    1. The work done by the gas on its surroundings.
    2. The heat flow to or from the gas.
    3. The change in internal energy of the gas.

Questions for Thermodynamics (1)

IDEAL GAS LAW

  1. Draw three P-V graphs for a an ideal gas (with number of molecules constant) at three different temperatures.
  2. Draw three P-T graphs for a an ideal gas (with number of molecules constant) at three different volumes.
  3. Draw three P-T graphs for a an ideal gas (with number of molecules constant) at three different volumes.
  4. On the three-dimensional axes of P, V, and T, draw a surface reflecting all the points that an ideal gas (with number of molecules constant) can exist at.
    • [hint : this will be easier to imagine with P axis positive upwards, V axis positive sideways, and T axis positive backwards]
    • [hint : the projection of the 3D surface onto the P-V plane for three different T values is the graph produced in question 1. same goes with corresponding variables in questions 2 and 3.]

FIRST LAW OF THERMODYNAMICS

  1. 5g of a gas with heat capacity 0.9J/K is kept at a constant volume, and its temperature falls from 293K to 273K. Assuming that it is ideal, what is its change in internal energy?
  2. 02.g of a gas with heat capacity 1.2kJ/K and molar mass 20g/mol is placed in a container of volume 0.5L and initial temperature 300K. It is allowed to expand to a volume of 1L with its temperature held constant. Afterwards, it is cooled to 250K with its volume held constant.
    1. How does the speed of the molecules in the gas change as it expands?
    2. How does the speed of the molecules in the gas change as it cools?
    3. What is the net change in the internal energy of the gas after the entire process?