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terminal-velocity

๐Ÿš€ Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium (like air or water) prevents further acceleration. This concept is crucial in understanding how objects behave when they fall through a fluid, influenced by factors such as viscosity and flow characteristics of the fluid. Viscosity is a measure of a fluid's resistance to deformation or flow, and it plays a significant role in determining the terminal velocity of an object. When an object falls through a viscous fluid, it experiences a drag force that opposes its motion, and at terminal velocity, the drag force equals the gravitational force acting on the object, resulting in no net acceleration.

Theory Explanation

Understanding Viscosity

Viscosity is a measure of a fluid's resistance to flow. It describes how thick or sticky a fluid is. For example, honey has a higher viscosity than water, meaning it flows more slowly. The viscosity of a fluid affects how quickly an object can fall through it. Higher viscosity means more resistance and a lower terminal velocity for falling objects.

\[ \eta = \frac{F}{A \cdot \frac{du}{dy}} \]
Calculating Terminal Velocity

The terminal velocity (v_t) can be calculated using the equation: v_t = (2mg)/(ฯC_dA), where m is the mass of the object, g is the acceleration due to gravity, ฯ is the density of the fluid, C_d is the drag coefficient, and A is the cross-sectional area of the object. This equation shows how terminal velocity depends on the object's properties and the fluid's characteristics.

\[ v_t = \frac{2mg}{\rho C_d A} \]
Equilibrium of Forces

At terminal velocity, the forces acting on the object are balanced. The gravitational force (weight) acting downwards is equal to the drag force acting upwards. This balance of forces is what allows the object to fall at a constant speed without accelerating further.

\[ mg = F_d \]

Key Points

  • ๐ŸŽฏ Terminal velocity is reached when the drag force equals the gravitational force.
  • ๐ŸŽฏ Viscosity affects the terminal velocity; higher viscosity results in lower terminal velocity.
  • ๐ŸŽฏ The terminal velocity formula incorporates mass, fluid density, drag coefficient, and cross-sectional area.

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Examples:💡

A sphere of radius 0.1 m and mass 0.5 kg is dropped in a fluid with a density of 1000 kg/mยณ and a drag coefficient of 0.5. Calculate the terminal velocity.

Solution:

Step 1: Identify the known values: m = 0.5 kg, g = 9.81 m/sยฒ, ฯ = 1000 kg/mยณ, C_d = 0.5, A = ฯ€(0.1)ยฒ = 0.0314 mยฒ.

\[ A = \pi r^2 = \pi (0.1)^2 = 0.0314 m^2 \]

Step 2: Substitute the values into the terminal velocity formula: v_t = (2mg)/(ฯC_dA).

\[ v_t = \frac{2 \cdot 0.5 \cdot 9.81}{1000 \cdot 0.5 \cdot 0.0314} = 0.313 m/s \]

Step 3: Calculate the terminal velocity: v_t = 0.313 m/s.

Common Mistakes

  • Mistake: Confusing viscosity with density; students often think they are the same.

    Correction: Remember that viscosity measures a fluid's resistance to flow, while density measures mass per unit volume.

  • Mistake: Forgetting to balance forces when calculating terminal velocity.

    Correction: Always set the gravitational force equal to the drag force to find terminal velocity.

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