Derive an expression for the vertical position of a projectile in terms of its horizontal position. This sometimes called parametric equation or path equation for the projectile. I don't know how to get it please help. Thank you.
Y = Vo*sinA*t -(g/2)t^2 and
X = Vo*cosA*t leads to
Y = Vo*sinA*X/(VocosA)
(g/2)*X^2/[Vo^2*cos^2A]
= X*tanA -(g/2)*X^2/[Vo^2*cos^2A]
A is the launch angle and V is the initial velocity.
The equation does not work for a vertical firing, since X = 0 in that case.
To derive the expression for the vertical position of a projectile in terms of its horizontal position, we can use the basic principles of projectile motion.
We start by considering the vertical motion of the projectile. The key concept is that the only force acting vertically on the projectile is gravity, which causes it to accelerate downwards at a constant rate. This means the motion follows a constant acceleration equation.
Let's assume that the initial velocity of the projectile is v₀ at an angle θ with respect to the horizontal. We can decompose this initial velocity into its horizontal and vertical components. The horizontal component, v₀x, remains constant throughout the motion. The vertical component, v₀y, changes due to the acceleration caused by gravity.
The equations for the components of the velocity are:
v₀x = v₀ * cos(θ)
v₀y = v₀ * sin(θ) - g * t
where g is the acceleration due to gravity and t is the time in seconds.
Now, let's consider the vertical position of the projectile, denoted as y. The vertical displacement can be described using the kinematic equation:
y = y₀ + v₀yt - (1/2)g*t²
where y₀ is the initial vertical position.
Since we are interested in expressing the vertical position in terms of the horizontal position, denoted as x, we need to relate the two variables.
The horizontal displacement, x, can be defined as:
x = v₀t
Rearranging this equation, we get:
t = x / v₀
Substituting this value of t into the equation for y, we have:
y = y₀ + (v₀ * sin(θ) - g * (x / v₀)) * (x / v₀) - (1/2)g * (x / v₀)²
Simplifying this expression, we get the final equation for the vertical position of the projectile in terms of its horizontal position:
y = y₀ + x * tan(θ) - (g * x²) / (2 * v₀² * cos²(θ))
Therefore, the derived expression for the vertical position of a projectile in terms of its horizontal position is:
y = y₀ + x * tan(θ) - (g * x²) / (2 * v₀² * cos²(θ))
This equation gives the vertical position as a function of the horizontal position, initial vertical position, initial velocity, angle of projection, and acceleration due to gravity.