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A stone in free fall covers distances $h_1$, $h_2$, $h_3$ in successive 5-second intervals. The correct relation is:
Distances in successive equal time intervals from rest are in the ratio of odd numbers: $1:3:5:7\ldots$
So $h_1 : h_2 : h_3 = 1 : 3 : 5$, i.e., $\mathbf{h_1 = \dfrac{h_2}{3} = \dfrac{h_3}{5}}$
Density ($\rho$) is given by: $\rho = \frac{M}{L^3}$
The relative error formula for density is:
$\frac{\Delta \rho}{\rho} \times 100 = \frac{\Delta M}{M} \times 100 + 3 \left( \frac{\Delta L}{L} \times 100 \right)$
Substitute the given values:
$\text{Max Error} = 1.5\% + 3(1\%) = 4.5\%$
Sharks belong to which class of vertebrates?
Class Chondrichthyes includes cartilaginous fish (sharks, rays). Osteichthyes are bony fish. Echinodermata are starfish etc.; Urochordata are tunicates.
Average time between collisions $\tau$ is given by $\tau = \frac{\lambda}{v_{rms}}$, where $\lambda$ is the mean free path.
1. $\lambda \propto \frac{1}{n} \propto V$ (where $n$ is number density).
2. $v_{rms} \propto \sqrt{T}$.
3. For an adiabatic process, $TV^{\gamma-1} = \text{constant}$, so $T \propto V^{1-\gamma}$ and $\sqrt{T} \propto V^{\frac{1-\gamma}{2}}$.
Therefore, $\tau \propto \frac{V}{V^{\frac{1-\gamma}{2}}} = V^{1 - \frac{1-\gamma}{2}} = V^{\frac{2-1+\gamma}{2}} = V^{\frac{\gamma+1}{2}}$.
Hence, $q = \frac{\gamma+1}{2}$.
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