projectile is fired from the ground s=0 ft with initial speed of v=800 ft/sec at an angle of elevation 30 degrees

State the vector function the projectile follow,projectile initial velocity vector,vector function,projectile speed function

To find the vector function of the projectile and the speed function, we need to break down the initial velocity vector into its horizontal and vertical components.

Given:
Initial speed (magnitude of velocity), v = 800 ft/sec
Angle of elevation, θ = 30 degrees

We can find the horizontal and vertical components using basic trigonometry:

The horizontal component of the initial velocity vector, Vx, can be found using the cosine of the angle of elevation:
Vx = v * cos(θ)

The vertical component of the initial velocity vector, Vy, can be found using the sine of the angle of elevation:
Vy = v * sin(θ)

1. Vector Function:
The vector function describes the position of the projectile as a function of time. Since we are given the initial position as 0 ft from the ground, the vector function can be expressed as:

r(t) = <x(t), y(t)>

Where:
x(t) denotes the horizontal position at time t
y(t) denotes the vertical position at time t

2. Projectile Initial Velocity Vector:
The projectile's initial velocity vector can be expressed as a combination of its horizontal and vertical components:

v0 = <Vx, Vy> = <v * cos(θ), v * sin(θ)>

3. Speed Function:
The speed of the projectile at any given time t can be calculated by finding the magnitude of the velocity vector:

v(t) = |v(t)| = √(Vx^2 + Vy^2)

Using the given values, we can substitute the respective formulas to obtain the final forms of the vector function and speed function depending on the variables used in these equations.