When a garden hose with an output diameter of 20 mm is directed straight upward, the stream of water rises to a height of 0.13m .
you then use your thumb to partially cover the output opening so that its diameter is reduced to 10 mm.

how high does the water rise now?
ignore drag and assume that the smaller opening you create with your thumb is circular.

y₂= 2.08  m

Explanation:

This problem should use the fluid continuity equation and kinematics to find the speeds.

Let's start by finding the initial velocity (vo) so that the water reaches the height (y1 = 0.13 m), with kinematics

    vf² = v₀² - 2 g y₁

    0 = v₀² - 2 g y₁

    v₀ = √ 2g y₁

    v₀ = √ (2 9.8 0.13)

    v₀ = 1,596 m / s

Now let's use the fluid continuity equation

   A₁ v₁ = A₂ v₂

The initial speed the speed with the largest diameter

   v₁ = v₀ = 1,596 m / s

Let's look for the areas

   r = d / 2

   r₁ = 20/2 = 10 mm = 10 10⁻³m

   r₂ = 10/2 = 5 mm = 5 10⁻³ m

   A = pi R²

   A₁ = π (10 10⁻³)²

   A₁ = π 10⁻⁴  m²

   A₂ = π5 10⁻³)²

   A₂ = π 25 10⁻⁶ m²2

We cleared the exit speed (v2) by reducing the diameter (d2)

    v₂ = v₁ A₁ / A₂

    v₂ = 1,596 π 10⁻⁴ / π 25 10⁻⁶

    v₂ = 6.38 m / s

Now we find the height for this speed again

   vf² = v₂² - 2g y₂

   0 = v₂² - 2 g y₂

  y₂ = v₂² / 2g

 y₂= 6.38² / (2  9.8)

 y₂= 2.08  m

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