3) A shell is fired from ground level with a muzzle speed of 350 ft / s and elevation and angle
of 60 . Find
(a) parametric equation for the shell’s trajectory
(b) the maximum height reached by the shell
(c) the horizontal distance traveled by the shell
(d) the speed of the shell at impact.
See Wed, 1:58AM post.
To find the answers to these questions, we can use kinematic equations and trigonometry. Let's break down each question one by one:
(a) Parametric equations for the shell's trajectory:
To derive the parametric equations, we need to consider the horizontal and vertical components separately.
Let's assume that the initial position of the shell is at (x0, y0) = (0, 0).
For the horizontal component:
We know that the horizontal velocity remains constant and is given by:
Vx = V₀ * cos(θ)
For the vertical component:
We know that the vertical position is determined by the initial vertical velocity and the acceleration due to gravity:
Vy = V₀ * sin(θ) - g * t
where g is the acceleration due to gravity, and t is the time.
We can express the x and y coordinates as functions of time (t):
x(t) = V₀ * cos(θ) * t
y(t) = V₀ * sin(θ) * t - (1/2) * g * t^2
So, the parametric equations for the shell's trajectory are:
x(t) = V₀ * cos(θ) * t
y(t) = V₀ * sin(θ) * t - (1/2) * g * t^2
(b) Maximum height reached by the shell:
The maximum height is achieved when the vertical velocity, Vy, becomes zero.
To find the time at which this occurs, we can set Vy = 0 and solve for t:
V₀ * sin(θ) - g * t = 0
t = V₀ * sin(θ) / g
Substituting this value of t into the equation for y(t), we can find the maximum height reached by the shell.
(c) Horizontal distance traveled by the shell:
The time of flight, T, is the total time the shell is in the air. To find it, we need to determine when the shell hits the ground, i.e., when y = 0.
Setting y(t) = 0 and solving for t will give us the time of flight.
Substituting this value of t into the equation for x(t), we can find the horizontal distance traveled by the shell.
(d) Speed of the shell at impact:
The speed of the shell at impact is the magnitude of the final velocity. We can find it by calculating the horizontal component of the final velocity (Vx) and the vertical component of the final velocity (Vy) and then using the Pythagorean theorem.