A ball is projected in still air. With respect to the ball the streamlines appear as shown in the figure. If speed of air passing through the region 1 and 2 are \( v_1 \) and \( v_2 \), respectively and the respective pressures, \( P_1 \) and \( P_2 \), respectively, then
Show Hint
Remember, in a fluid flow, when the velocity increases, the pressure decreases, and vice versa. This is known as Bernoulli’s principle.
According to Bernoulli’s principle, for a steady flow of an incompressible fluid, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant. This can be expressed as: \[ P + \frac{1}{2} \rho v^2 + \rho gh = {constant} \] For horizontal flow (ignoring potential energy), the equation simplifies to: \[ P + \frac{1}{2} \rho v^2 = {constant} \] When the speed of air increases, the pressure decreases to maintain the constant value. From the figure, we see that as the air passes through the narrow part of the streamline (region 1), the speed of air \( v_1 \) increases, and the pressure \( P_1 \) decreases. In the wider part (region 2), the air speed \( v_2 \) is slower, and the pressure \( P_2 \) is higher. Thus, \( v_1<v_2 \) and \( P_1>P_2 \). Therefore, the correct answer is (E).
