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keisanHigh-precision calculations for everyday life and work

Velocity of a sphere in a horizontal projectile motion subject to air resistance

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Find the velocity of the sphere after an arbitrary time
Calculate the velocity of a sphere at any given time after a horizontal projectile launch subject to air resistance. Note that the sphere’s surface irregularities and rotation are not taken into account.
*Vertical velocity changes are not calculated.
T Absolute Temperature (K)
(K)
p Atmospheric pressure (hPa)
(hPa)
r Radius (m)
(m)
m Mass (kg)
(kg)
t: Arbitrary elapsed time (s)
(s)
v0 Initial velocity (m/s)
(m/s)
Velocity v at time t (m/s) (km/h)
Aerodynamic drag coefficient k (1/m)
Air Density ρ (kg/m³)
The default answer is the speed of a hardball thrown at 162 km/h under 1 atm and 0°C when it reaches the catcher (0.4 seconds later). (In this case, the critical Reynolds number is reached, so the actual speed is even faster.)
The drag coefficient always takes a value of 0 or less; the larger the value in the negative direction, the greater the air resistance.
The drag coefficient is set to 0.4.

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