Projectile Motion
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Projectile Motion
Trajectories of a mass thrown at an angle of 70°:
without drag
with Stokes drag
with Newton drag

Projectile motion is a form of motion experienced by a launched object. Ballistics (Greek: ?, romanizedba'llein, lit.'to throw') is the science of dynamics that deals with the flight, behavior and effects of projectiles, especially bullets, unguided bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.

## Kinematic quantities of projectile motion

In a uniform gravitational field without air resistance, the horizontal and vertical components of velocity are independent from each other. Galileo Galilei dubbed this principle compound motion' in 1638[1] and used it to prove that the trajectory of a projectile is a parabola.[2] The projectile's horizontal and vertical displacement as a function of time are:

${\displaystyle x(t)=v_{0}t\cos(\theta )}$,
${\displaystyle y(t)=v_{0}t\sin(\theta )-{\frac {1}{2}}gt^{2}}$,

where v0 is the initial speed, θ is the angle of the initial velocity with respect to the horizontal direction, and g is the downward gravitational acceleration.

## Trajectory of a projectile with air resistance

Trajectories of a projectile with air drag and varying initial velocities

Air resistance creates a force that depends on the projectile's speed through the medium.[3] The speed-dependence of the friction force is linear (${\displaystyle f(v)\propto v}$) at very low speeds (Stokes drag) and quadratic (${\displaystyle f(v)\propto v^{2}}$) at larger speeds (Newton drag).[4] The transition between these behaviours is determined by the Reynolds number, which depends on speed, object size and kinematic viscosity of the medium. For Reynolds numbers below about 1000, the dependence is linear, above it becomes quadratic. Qualitatively, the speed approaches a terminal velocity ${\displaystyle v_{\infty }}$ that depends on the drag and the particle's mass. The trajectory has a limited horizontal range, becomes vertically downward near this vertical asymptote, and reaches its maximum height lower and sooner than in the case of no air resistance.[5]

### Trajectory of a projectile with Stokes drag

Stokes drag, where ${\displaystyle \mathbf {F_{air}} =b\mathbf {v} }$, only applies at very low speed in air, and is thus not the typical case for projectiles. However, the linear dependence of ${\displaystyle F_{\mathrm {air} }}$ on ${\displaystyle v}$ causes a very simple differential equation of motion

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\begin{pmatrix}v_{x}\\v_{y}\end{pmatrix}}={\begin{pmatrix}-\mu \,v_{x}\\-g-\mu \,v_{y}\end{pmatrix}}}$

in which the two cartesian components become completely independent, and thus easier to solve.[6] The solution in this approximation may be expressed in closed form as:[7]

${\displaystyle x(t)=v_{0}\tau \cos \theta (1-e^{-t/\tau })}$
${\displaystyle y(t)=\tau (v_{0}\sin \theta +v_{\infty })(1-e^{-t/\tau })-v_{\infty }t}$,

where ${\displaystyle \tau =m/b}$, and the terminal velocity is ${\displaystyle v_{\infty }=g\tau }$.

### Trajectory of a projectile with Newton drag

The most typical case of air resistance, for the case of Reynolds numbers above about 1000 is Newton drag with a drag force proportional to the speed squared, ${\displaystyle F_{\mathrm {air} }=-kv^{2}}$, and the terminal velocity is ${\displaystyle v_{\infty }={\sqrt {mg/k}}}$. In air, which has a kinematic viscosity around ${\displaystyle 0.15\,\mathrm {cm^{2}/s} }$, this means that the product of speed and diameter must be more than about ${\displaystyle 0.015\,\mathrm {m^{2}/s} }$. The general case cannot be solved analytically, but an exact result can be found for vertical downward motion:[8][3]

${\displaystyle y(t)={\frac {v_{\infty }^{2}}{g}}\ln \left(\cosh {\frac {gt}{v_{\infty }}}\right)}$.

## Lofted trajectory

Lofted trajectories of North Korean missiles Hwasong-14 and Hwasong-15

A special case of a ballistic trajectory for a rocket is a lofted trajectory, a trajectory with an apogee greater than the minimum-energy trajectory to the same range. In other words, the rocket travels higher and by doing so it uses more energy to get to the same landing point. This may be done for various reasons such as increasing distance to the horizon to give greater viewing/communication range or for changing the angle with which a missile will impact on landing. Lofted trajectories are sometimes used in both missile rocketry and in spaceflight.[9]

## References

1. ^ Galileo Galilei, Two New Sciences, Leiden, 1638, p. 249.
2. ^ Nolte, David D., Galileo Unbound (Oxford University Press, 2018) pp. 39-63.
3. ^ a b Taylor, John R. (2005). Classical Mechanics. Mill Valley, California. pp. 61-64. ISBN 978-1-891389-22-1.
4. ^ Stephen T. Thornton; Jerry B. Marion (2007). Classical Dynamics of Particles and Systems. Brooks/Cole. p. 59. ISBN 978-0-495-55610-7.
5. ^ Taylor, John R. (2005). Classical Mechanics. Mill Valley, California. pp. 55, 64-65. ISBN 978-1-891389-22-1.
6. ^ Atam P. Arya; Atam Parkash Arya (September 1997). Introduction to Classical Mechanics. Prentice Hall Internat. p. 227. ISBN 978-0-13-906686-3.
7. ^ Taylor, John R. (2005). Classical Mechanics. Mill Valley, California. p. 54. ISBN 978-1-891389-22-1.
8. ^ Walter Greiner (2004). Classical Mechanics: Point Particles and Relativity. Springer Science & Business Media. p. 181. ISBN 0-387-95586-0.
9. ^