Projectile motion

Projectile motion is one of the traditional branches of classical mechanics, with applications to ballistics. A projectile is any body that is given an initial velocity and then follows a path determined by the effect of the gravitational acceleration and by air resistance. For example, a thrown football, an object dropped from an airplane, or a bullet shot from a gun are all examples of projectiles.

The path followed by a projectile is called its trajectory.

The initial velocity v₀ can be written as

$v_{0}=v_{0x}\mathbf {i} +v_{0y}\mathbf {j}$

The components v_0x and v_0y can be found if the angle θ₀ is known:

$v_{0x}=v_{0}\cos \theta _{0}$

and

$v_{0y}=v_{0}\sin \theta _{0}$

The horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.

Since there is no acceleration in the horizontal direction, the horizontal component of the velocity remains unchanged throughout the motion, as shown in figure.

The horizontal displacement

$x-x_{0}$

from an initial position x₀ is given by the equation

$x-x_{0}=v_{0x}t+{\begin{matrix}{\frac {1}{2}}\end{matrix}}at^{2}$,

with a = 0 and $v_{0x}=v_{0}\cos \theta _{0}$. Thus

$x-x_{0}=(v_{0}\cos \theta _{0})t$.

The vertical motion is the motion of a particle in free fall. Equations for free fall apply. For example, $y-y_{0}=(v_{0}\sin \theta _{0})t-{\begin{matrix}{\frac {1}{2}}\end{matrix}}gt^{2}$. Other useful equations for the vertical y-axis are $v_{y}=v_{0}\sin \theta _{0}-gt$, and $v_{y}^{2}=(v_{0}\sin \theta _{0})^{2}-2g(y-y_{0})$.

Eliminating t between the following two equations, $x-x_{0}=(v_{0}\cos \theta _{0})t$ and $y-y_{0}=(v_{0}\sin \theta _{0})t-{\begin{matrix}{\frac {1}{2}}\end{matrix}}gt^{2}$, we obtain the equation of the path (the trajectory) of the projectile:

$y=x\tan(\theta )-xg/2v_{0}cos(\theta ).$

Helpful equations

Time to reach the maximum height

$t={v_{0y} \over g}$

Time to reach ground

$t={2v_{0y} \over g}$

Range of projectile

${r={\frac {v^{2}}{g}}\sin(2\theta )}\,$

Maximum height

$y_{max}={{v_{0y}}^{2} \over 2g}$

Parabolic trajectory

Since g, θ₀, and v₀ are constants, the above equation is of the form

$y=ax+bx^{2}$,

in which a and b are constants. This is the equation of a parabola, so the path is parabolic.

The horizontal range R of the projectile is the horizontal distance the projectile has traveled when it returns to its initial height:

Note that R has its maximum value when $\sin 2\theta _{0}=1$, which corresponds to $2\theta _{0}=90^{\circ }$ or $\theta _{0}=45^{\circ }$.

See also

Trajectory

External links

• A Java simulation of projectile motion, including first-order air resistance
• MIT Video Lecture on 3-D projectile motion by Walter Lewin