Physics: Projectile Motion

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May 24, 2022, 3:22:09 PM

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MichaelBartmess

Dec 26, 2014, 11:16:45 PM

Component Motion

Motion With Launch Angle

Ballistic Trajectory

Component Motion

The basic projectile motion can be decomposed for effective analysis into an x-component parallel to the local horizontal plane and a y-component, normal to that same plane. The y-component on the Earth can be thought of as parallel to the radius of the Earth. In the ideal ballistic projectile model:

the y-component of acceleration is due solely to the acceleration due to gravity² ; [ See the calculator button labeled "Y-acceleration of projectile"].
the x-component of acceleration is zero³, since we are ignoring other forces such as air resistance.

X-Component

We start with the basic kinematic equation for displacement applied in the x-direction:

`v_x = v_"0x" + a_x*t` ⁴ and setting `a_x` to zero as just discussed, we get `v_x = v_"0x"`. ⁵ [ See the calculator button labeled: "X-Velocity - constant acceleration" ]

This basically tells us that, neglecting air resistance, the horizontal velocity will remain constant at the initial x-component velocity. And from this we get the displacement in the x-direction as:

`x = x_0 + v_"0x" * t` ⁶ [ See the calculator button labelled: "X-Displacement - Constant Velocity" ]

So, the equations of x-component motion for an ideal projectile are:

`x = x_0 + v_"0x"*t` ⁷
`v_x = v_"0x"` ⁸
`a_x = 0` ⁹

If we start at the origin where x= 0, then the equation for displacement becomes simple: `x = v_"0x"*t`

Y-Component

Next we look at the same basic kinematic equations as applied to the y-component of projectile motion:

`v_y = v_"0y" + a_y*t` ⁴ , which becomes `v_y = v_"0y" - g*t`¹⁰ when we substitute the fact that the acceleration near the surface of the Earth is -g.

And the displacement in the y-direction is derived analogously to the displacement in x-direction as above.

/attachments/4269391e-8d55-11e4-a9fb-bc764e2038f2/Vector x-y components.png `y = y_0 + v_"0y" * t -1/2 * g*t^2` ¹¹ ¹² [ See the calculator button labelled: "Y-Displacement - Constant Acceleration" ]

Component Motion with Launch Angle

The projectile motion has a launch angle. Typically we represent the path of motion as passing through the coordinate system's origin at time, t = 0. At this instant the velocity vector is at the launch angle, `phi`, to the x-axis.

If the magnitude of the initial velocity is `v_0`, then the initial velocity at time, t = 0, can be represented by x and y components as:

`v_"0x" = v_0 * cos(phi)` ¹³ [See the calculator button labeled "Ballistic Initial X Velocity"]
`v_"0y" = v_0 * sin(phi)` ¹⁴ [See the calculator button labeled "Ballistic Initial Y Velocity"]

Using the equation reflected above for x and y-displacement and x and y velocity, we can substitute these two initial velocity components (`v_"0x", v_"0y"`) to get the component displacement and velocity equations:

`x = (v_0*cos(phi_0) * t` ¹⁵ [ See calculator button labeled Displacement - x(`v_0`,t,`theta`) ]

`y = v_0 * sin(phi_0) *t - 1/2g*t^2` ¹⁶ [ See calculator button labeled Displacement - y(t,`theta, v_0`) ]

`v_x = v_0*cos(phi_0)` ¹⁷ [ See calculator button labeled Ballistic X Velocity ]

`v_y = v_0 * sin(phi_0) - g*t` ¹⁸ [ See calculator button labeled Ballistic Y Velocity ]

If you know the initial velocity components, `v_x` and `v_y`, you can compute the launch angle, `alpha` ¹⁹, shown in Figure 1: `alpha = arctan(v_y/v_x)`

Projectile Trajectory

With the component definition of projectile motion we can derive the following information about the projectile and its trajectory:

The distance in the x-y plane at any time, t, is given by `r = sqrt(x^2 + y^2)` ²⁰, which can also be expressed in three dimensions as `r = sqrt(x^2 + y^2 + z^2)` [ See calculator button labeled

Similarly, the velocity at any time, t, can be computed from the velocity components at that time: `v = sqrt(v_x^2 + v_y^2)` ²¹ and in three dimensions as `v = sqrt(v_x^2 + v_y^2 + v_z^2)`

We can solve the equation for x-component displacement, `x = (v_0*cos(phi_0) * t`, for t: `t = x / (v_0 * cos(phi_0))`.

This allows us to substitute for t in the equation for y-component displacement and derive the equation for y in terms of x. This then gives us the trajectory equation:

`y = v_0 * sin(phi_0) *t - 1/2*g*t^2` [ See the calculator button labeled Displacement - y(t, `theta`, `v_0`) ]

`y = v_0 * sin(phi_0) * [x / (v_0 * cos(phi_0))] - 1/2g*[x / (v_0 * cos(phi_0))]^2`

`y = tan(phi_0)*x - (g*x^2) / (2* v_0^2 * cos^2(phi_0))` ²² [ See the calculator button labeled Displacement - y(x, `theta`, `v_0`) ]

It is important to note that since `v_0, phi_0, and g` are constants this equation takes the form of `y = b*x -c*x^2`, which is recognizable as the equation for a parabola. The simplified trajectory, ignoring all forces besides gravity, will always be a parabolic curve.

Since we can decompose the position or velocity vector into its components parallel to the x-y-z axes, we can compute the distance of the projectile from the origin: `|vecr|` = r_x^2 + r_y^2 + r_z^2`. [ See calculator button labeled Projectile Distance ]

We can also compute the ballistic projectile's maximum height above the launch point altitude: `H_"max" = (v_0 * sin(theta))^2 / (2*g)` [ See calculator button labeled Ballistic Max Height ]

[Figure-1] Initial velocity of parabolic throwing
Source: Wikipedia / Fizped (modified to include motion equation)
URL: http://en.wikipedia.org/wiki/Projectile_motion#mediaviewer/File:Ferde_hajitas1.svg

Physics: Projectile Motion

Component Motion

X-Component

Y-Component

Component Motion with Launch Angle

Projectile Trajectory

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