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21 cze 2024 · The drag is given by the drag equation: \(\LARGE D=\frac{\mathit{Cd}\cdot\rho AV_{t}^{2}}{2}\) where rho ( \(\bf\rho\) ) is the gas density, Cd is the drag coefficient which characterizes the effects of shape of the ball, A is the cross-sectional area of the ball, and V t is the terminal velocity.
Projectile Motion with Air Resistance. Suppose that a projectile of mass is launched, at , from ground level (in a flat plain), making an angle to the horizontal. Suppose, further, that, in addition to the force of gravity, the projectile is subject to an air resistance force which acts in the opposite direction to its instantaneous direction ...
In our study of projectile motion, we assumed that air-resistance effects are negli-gibly small. But in fact air resistance (often called air drag, or simply drag) has a major effect on the motion of many objects, including tennis balls, bicycle riders, and airplanes.
Construct a mathematical model of an object shot into the air and subject to the forces of drag and gravity. Variables and Parameters#
The magnitude of the drag force is characterized by the dimensionless drag coefficient $C_{\rm D}$, given by \[ C_{\rm D} = \frac{F_{\rm D}}{\frac{1}{2} \rho A v^2}, \] where $\rho$ is the density of the fluid (in this case, air), $A=(1/4)\pi D^{2}$ is the cross-sectional area of the object, and $v$ is the object's speed.
Linear drag. The first example I will consider here is the simplest, which is horizontal motion with linear drag. (You can realize this experimentally by immersing a frictionless cart on a surface in some viscous liquid, like honey or molasses.)
In particular, you can solve these equations by specifying the initial position $\mathbf x(0) = (x(0), y(0)$ and the initial velocity $\mathbf v(0) = (v_x(0), v_y(0)) = (v(0)\cos\theta, v(0)\sin\theta)$ where $\theta$ is the initial angle at which the projectile is launched.
