Water is flowing at a speed of 1.5 m s$$^{-1}$$ through a horizontal tube of cross-sectional area $$10^{-2}$$ m$$^2$$ and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm, the minimum force that you must exert should be (density of water = $$10^3$$ kg m$$^{-3}$$)

To solve this problem, we need to find the minimum force required to stop the flow of water using your palm. The water flows at a speed of 1.5 m/s through a tube with a cross-sectional area of $$10^{-2}$$ m², and it stops immediately upon hitting the palm. The density of water is $$10^3$$ kg/m³.

The force exerted by the palm must counteract the momentum of the incoming water per unit time. Force is defined as the rate of change of momentum. When the water hits the palm and stops, its velocity changes from 1.5 m/s to 0 m/s. The change in velocity ($$\Delta v$$) is therefore $$1.5 - 0 = 1.5$$ m/s.

First, we calculate the volume flow rate, which is the volume of water passing through the tube per second. This is given by the product of the cross-sectional area ($$A$$) and the flow velocity ($$v$$):

Volume flow rate = $$A \times v = 10^{-2} \, \text{m}^2 \times 1.5 \, \text{m/s} = 0.015 \, \text{m}^3/\text{s}$$.

Next, we find the mass flow rate, which is the mass of water flowing per second. This is obtained by multiplying the volume flow rate by the density of water ($$\rho$$):

Mass flow rate = $$\rho \times \text{volume flow rate} = 10^3 \, \text{kg/m}^3 \times 0.015 \, \text{m}^3/\text{s} = 15 \, \text{kg/s}$$.

The rate of change of momentum is equal to the mass flow rate multiplied by the change in velocity ($$\Delta v$$), since the water comes to a complete stop:

Rate of change of momentum = mass flow rate $$\times \Delta v = 15 \, \text{kg/s} \times 1.5 \, \text{m/s} = 22.5 \, \text{kg} \cdot \text{m/s}^2$$.

Since $$1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2$$, the force required is 22.5 N. This is the minimum force needed to stop the water flow, assuming an immediate stop with no other losses.

Comparing with the options: A. 33.7 N, B. 45 N, C. 15 N, D. 22.5 N. Hence, the correct answer is Option D.