Analyze the dynamics of an object reaching terminal velocity.

Analyze the dynamics of an object reaching terminal velocity.

When an object falls freely towards Earth, it is initially accelerated by gravity. However, as it gains speed, the resistance from air (air resistance or drag) also increases.

Stages of Falling:

  1. Initial Free Fall: In the initial stage, when the object is first released, the force of gravity dominates, and the object accelerates downwards at 9.8 m/s2 (acceleration due to gravity).

  2. Increasing Air Resistance: As the object speeds up, the air resistance also increases because it is proportional to the square of the velocity (Fdrag∝v2). This resistance pushes upwards, opposing the downward force of gravity.

  3. Reaching Terminal Velocity: Eventually, the force of air resistance becomes equal to the force of gravity. At this point, the net force on the object is zero, meaning there is no longer any acceleration. The object then falls at a constant velocity, known as terminal velocity.

The terminal velocity depends on several factors:

  • Shape and size of the object: A larger object with more surface area experiences more air resistance, which can lead to a lower terminal velocity.
  • Density of the air: Higher altitude or different atmospheric conditions can affect the density of air, influencing the drag force.
  • Mass of the object: Heavier objects experience a greater downward force, but this alone doesn’t determine terminal velocity; the drag force also depends on the shape and surface area.

For example:

  • A skydiver initially accelerates but eventually reaches terminal velocity around 53 m/s (about 120 mph) when falling belly-down.
  • A feather or a parachute, with a larger surface area, reaches terminal velocity at a much lower speed because it experiences more air resistance.

Mathematical Expression for Terminal Velocity:

The terminal velocity (vt) is reached when the drag force (Fdrag) equals the gravitational force (Fg):

Fdrag

1/2CdρAvt2=mg

Where:

  • Cd = drag coefficient (depends on shape),
  • ρ = air density,
  • = cross-sectional area,
  • vt= terminal velocity,
  • = mass of the object,
  • = acceleration due to gravity.

By rearranging the above equation, we can calculate the terminal velocity for any object.

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