Air ( γ = 1.4 ) is pumped to a pressure of 2 atm inside a motor...

The sudden bursting of a tyre is a fast process, hence it is considered an adiabatic expansion.

For an adiabatic process, the relationship between temperature and pressure is given by:

$T^{\gamma} P^{1-\gamma} = \text{constant}$

$\Rightarrow T_2 = T_1 \left( \dfrac{P_2}{P_1} \right)^{\dfrac{\gamma-1}{\gamma}}$

Given:

T_1 = 20^{}^{{}^{\circ}} \text{C} = 293 \text{ K}

$P_1 = 2 \text{ atm}$

$P_2 = 1 \text{ atm}$ (atmospheric pressure)

$\gamma = 1.4$

Substituting the values into the equation:

$T_2 = 293 \left( \dfrac{1}{2} \right)^{\dfrac{1.4-1}{1.4}}$

$T_2 = 293 \left( \dfrac{1}{2} \right)^{\dfrac{0.4}{1.4}} = 293 \left( 0.5 \right)^{\dfrac{2}{7}}$

Using the approximation $(0.5)^{\dfrac{2}{7}} \approx 0.8203$:

$T_2 \approx 293 \times 0.8203 \approx 240 \text{ K}$