Geotechnical Engineering: Understanding the Foundation of Infrastructure

 Horizontal motion

The horizontal component of the velocity vx remains unchanged throughout the motion.

vx=v0x=v0cosϑ0=const [

m

s

], where:
v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis

---

Horizontal displacement

Horizontal displacement is described by the formula:
x(t)=x0+v0xt
x(t)=x0+(v0cosϑ0)t [m], where:
x0 - initial position
v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis
t - time

---

Vertical motion

The vertical component of the velocity vy increases linearly, because the acceleration due to gravity g is constant.

vy=v0y−gt=v0sinϑ0−gt [

m

s

], where:
v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis
t - time
g - acceleration due to gravity

---

Vertical displacement

Vertical displacement is described by the formula:
y(t)=y0+v0yt−

gt2

2

y(t)=y0+(v0sinϑ0)t−

gt2

2

 [m], where:
x0 - initial position
v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis
t - time
g - acceleration due to gravity

---

Trajectory

If we eliminate t between x(t) and y(t) equations we will obtain the following equation:
y=(tgϑ0)x−

gx2

2(v0cosϑ0)2

v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis
g - acceleration due to gravity

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Range of a projectile time dependent

Assuming a flat Earth with a uniform gravity field, and no air resistance, a projectile launched with specific initial conditions will have a predictable range. This range time dependent is the total horizontal distance traveled by the projectile and can be described by the formula:

R(t)=(v0cosϑ0)t [m], where:
v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis

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Range of a projectile time independent

Assuming a flat Earth with a uniform gravity field, and no air resistance, a projectile launched with specific initial conditions will have a predictable range. This range time independent is the total horizontal distance traveled by the projectile and can be described by the formula:
R=

v02

g

sin(2ϑ0) [m], where:
v0 - initial velocity
ϑ0 - launch angle - the angle between v0 and positive direction of the x axis
g - acceleration due to gravity