The terminal velocity of an object dropped from a skyscraper depends on various factors, including the size, shape, and mass of the object, as well as the air density and viscosity.
Assuming we're considering a typical coin, such as a penny or a quarter, we can estimate its terminal velocity based on average values.
The terminal velocity occurs when the gravitational force pulling the object downward is balanced by the drag force exerted by the air resisting the object's motion. At this point, the object no longer accelerates and falls at a constant velocity.
For a coin, the terminal velocity typically ranges from 10 to 60 miles per hour (16 to 96 kilometers per hour). However, let's calculate a rough estimate based on average values.
The drag force experienced by an object falling through the air can be approximated using the drag equation:
F_drag = (1/2) * ρ * A * C_d * v²
where:
- F_drag is the drag force,
- ρ is the air density,
- A is the projected area of the coin (assuming it falls flat),
- C_d is the drag coefficient, and
- v is the velocity of the coin relative to the air.
Given the uncertainty about the specific coin and its dimensions, let's make some assumptions:
- ρ ≈ 1.2 kg/m³ (average air density near the Earth's surface)
- A ≈ 0.001 m² (estimated projected area of a typical coin)
- C_d ≈ 0.5 (average drag coefficient for a flat object)
- Convert terminal velocity to m/s: 60 mph ≈ 27 m/s
Rearranging the drag equation to solve for v:
v = sqrt((2 * F_drag) / (ρ * A * C_d))
Substituting the assumed values:
v = sqrt((2 * F_drag) / (1.2 * 0.001 * 0.5)) 27 = sqrt((2 * F_drag) / (1.2 * 0.001 * 0.5))
Squaring both sides:
729 = (2 * F_drag) / (1.2 * 0.001 * 0.5) F_drag = 729 * 1.2 * 0.001 * 0.5 / 2
Simplifying:
F_drag ≈ 0.437 N
Now, we can solve for the terminal velocity:
v = sqrt((2 * F_drag) / (ρ * A * C_d)) v = sqrt((2 * 0.437) / (1.2 * 0.001 * 0.5)) v ≈ sqrt(0.724 / 0.0003) v ≈ sqrt(2413.33) v ≈ 49.1 m/s
Therefore, based on these estimations, the terminal velocity of a coin dropped from the top of a skyscraper would be approximately 49.1 m/s. Please note that this is a rough estimate, and the actual terminal velocity can vary depending on several factors and the specific characteristics of the coin.