How does gravity relate to functions, specifically to s(t)= -1/2 gt^2 + vt + h.
Provide a brief history of the projectile and its use in functions.
projectile with no air friction and constant acceleration down of g
Vertical problem:
initial height = Hi
initial velocity component UPWARD = Vi
velocity upward = v
height = y
then
acceleration =-g which is about -9.81 m/s^2 on earth
v = Vi - g t
y = Hi + Vi t -(1/2)g t^2
That is all about up and down
There are no forces horizontal so the initial horizontal speed is forever until the crash.
call that speed u
then horizontal distance x = u t
===================
IF
you fire missile with speed S and angle up A
Then
Vi = S sin A
and
u = S cos A
the end :)
gravity provides a force of attraction between masses
a force acting on a mass causes it to accelerate
the function [s(t)= -1/2 gt^2 + vt + h] describes the height (s) relative to time (t) of a mass (object) in freefall
... g is the gravitational acceleration
... v is the initial vertical velocity
... h is the initial height
due to gravity, projectiles follow a parabolic
trajectory
... the horizontal velocity is constant
... while the vertical velocity changes due to gravity
Gravity is a fundamental force in physics that determines the motion of objects in relation to a massive body, such as the Earth. It gives rise to a variety of phenomena, including the motion of projectiles. To understand how gravity relates to functions like s(t) = -1/2gt^2 + vt + h, let's break it down:
In the given function, s(t) represents the position of an object (projectile) at a given time (t) in a vertical direction. The function contains three key elements:
- The first term, -1/2 gt^2, accounts for the effect of gravity. It represents the gravitational acceleration, g, acting on the projectile, causing it to accelerate downwards. The term is multiplied by -1/2 as gravity causes a deceleration in the upward direction.
- The second term, vt, represents the initial velocity, v, at which the object is launched. It signifies the horizontal motion, assuming no other external forces affect the projectile's horizontal speed.
- The third term, h, represents the initial height or displacement of the projectile from a reference point. It provides a vertical offset to the entire function.
To understand the historical context of projectiles and their use in functions, we can look back to ancient times. The ancient Greeks were amongst the first to study projectile motion. They recognized that when an object is thrown into the air, it follows a curved path known as a parabola.
The Greek polymath Archimedes furthered the understanding of projectile motion by introducing mathematical principles. He derived equations to describe the motion of projectiles and observed that the trajectory of a projectile depended on both its initial velocity and the angle at which it was launched.
Later, the study of projectiles evolved with the contributions of scientists like Galileo Galilei and Isaac Newton. Galileo's experiments and observations led him to formulate his laws of motion, which included the principle of inertia and the concept of parabolic projectile motion.
Isaac Newton's laws of motion, particularly his second law stating that force is proportional to mass times acceleration, significantly advanced the understanding of gravity's impact on projectiles. Newton's laws allowed for the development of mathematical models, such as the one mentioned earlier, to describe the motion of projectiles more accurately.
Today, projectile motion and related functions are widely used in fields such as physics, engineering, and ballistics. They help calculate various parameters, including range, maximum height, time of flight, and impact velocity, for objects launched into the air under the influence of gravity.
To better comprehend the relationship between gravity and projectile motion, studying the derivation and mathematical foundations of the equations involved can provide a clearer understanding.