Source: Wikipedia / Fizped (modified to include motion equation) We can also compute the ballistic projectile's maximum height above the launch point altitude: `H_"max" = (v_0 * sin(theta))^2 / (2*g)` 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`. The simplified trajectory, ignoring all forces besides gravity, will always be a parabolic curve. 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. This then gives us the trajectory equation:
![projectile motion physics calculator projectile motion physics calculator](https://www.physicstutorials.org/images/stories/example12last.png)
This allows us to substitute for t in the equation for y-component displacement and derive the equation for y in terms of x.
![projectile motion physics calculator projectile motion physics calculator](https://i.ytimg.com/vi/u8C0jp_61jE/maxresdefault.jpg)
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))`. 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)` 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 With the component definition of projectile motion we can derive the following information about the projectile and its trajectory: 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 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: 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: At this instant the velocity vector is at the launch angle, `phi`, to the x-axis. Typically we represent the path of motion as passing through the coordinate system's origin at time, t = 0. The projectile motion has a launch angle. `y = y_0 + v_"0y" * t -1/2 * g*t^2` 11 12 Component Motion with Launch Angle `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.Īnd the displacement in the y-direction is derived analogously to the displacement in x-direction as above. Next we look at the same basic kinematic equations as applied to the y-component of projectile motion: If we start at the origin where x= 0, then the equation for displacement becomes simple: `x = v_"0x"*t` Y-Component So, the equations of x-component motion for an ideal projectile are: And from this we get the displacement in the x-direction as: This basically tells us that, neglecting air resistance, the horizontal velocity will remain constant at the initial x-component velocity. `v_x = v_"0x" + a_x*t` 4 and setting `a_x` to zero as just discussed, we get `v_x = v_"0x"`. We start with the basic kinematic equation for displacement applied in the x-direction: the x-component of acceleration is zero 3, since we are ignoring other forces such as air resistance.the y-component of acceleration is due solely to the acceleration due to gravity 2.The y-component on the Earth can be thought of as parallel to the radius of the Earth. 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. velocity - the rate of change of the displacement of an object.trajectory - the ballistic path followed by a projectile.projectile - an object whose path of motion can be modeled using initial velocity and is affected by such physical phenomena as gravitational acceleration, air resistance, and the rotation of the planet.
![projectile motion physics calculator projectile motion physics calculator](https://i.ytimg.com/vi/1i-w4y55jS0/maxresdefault.jpg)
ballistics - the science of mechanics that deals with the launching, flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like 1.acceleration - the rate of change of velocity.