For incompressible fluids, the principle of continuity dictates that the flow rate (volume per unit time) is constant, expressed as:
\[A_1 v_1 = A_2 v_2\]
Where:
- \( A_1 = \pi r_1^2 \) represents the cross-sectional area at the wider end.
- \( A_2 = \pi r_2^2 \) represents the cross-sectional area at the narrower end.
- \( v_1 = 5 \, \text{m/s} \) is the velocity at the wider end.
- \( v_2 \) is the velocity at the narrower end.
Substituting the given values:
\[A_1 v_1 = A_2 v_2\]
\[\pi (2)^2 \times 5 = \pi (1)^2 \times v_2\]
\[4 \times 5 = 1 \times v_2\]
\[v_2 = 20 \, \text{m/s}\]
Therefore, the velocity of the water at the narrower end is \( 20 \, \text{m/s} \).