Projectile Motion and Linear Forces in Physics

Posted on May 19, 2024 in Physics

Positive and Negative Acceleration

Positive Acceleration (+a)Negative Acceleration (-a)
Positive Initial Velocity (+Vo)
  • Moving forward
  • Speeding up
  • Continues as long as there is acceleration
  • Moving forward
  • Slowing down until stopped
  • Then, speeding up backward
  • Continues as long as there is acceleration
Negative Initial Velocity (-Vo)
  • Moving backward
  • Slowing down until stopped
  • Then, speeding up forward
  • Continues as long as there is acceleration
  • Moving backward
  • Speeding up
  • Continues as long as there is acceleration

Acceleration Due to Gravity

  • Caused by all objects with mass
    • Newtonian: a force which is always attractive
    • Einsteinian: a curvature in spacetime

g=GM/r2

  • M is mass of the “gravity maker (kg)”
  • r is the distance to the centre of the “gravity maker”
  • G is Newtonian Gravitation Constant
    • (6.6710-11m3kg-1s-2)
  • At centre of body g=0

Projectile Motion

Projectile

  • Mass which moves under the influence of only gravity and/or drag (air resistance)
  • It does not have thrust
  • It is not on a constrained track (e.g., rollercoaster)

Our Projectiles

  • Are in vacuum only (no air resistance)
  • Are ground launched
  • Are not on a ramp, which would affect the launch angle
  • Impact at the same height at which they were launched (unless intercepted in flight)

Projectile

  • Maximum horizontal distance = Range (R)
  • Maximum vertical height = Apogee (A)

The Launch Conditions

  • At Launch:

(1)Vox=(Vo)(Cos(theta))

(2)Voy=(Vo)(Sin(theta))

The Velocities After Launch

  • There is no horizontal acceleration

(3) Vx(t)=(Vo)(Cos(theta))

(4) Vy(t)=(Vo)(Sin(theta))-(g)(t)

The Positions After Launch

  • The Horizontal Position

(5) x(t)=(Vo)(t)(Cos(theta))

  • The Vertical Position

(6) y(t)=(Vo)(t)(Sin(theta))-(1/2)(g)(t2)

The Apogee Time (tA)

(7) tA=(Vo)(Sin(theta)) / (g)

The Maximum Apogee Time (tAmax)

(8) tmax= Vo/g

The Apogee (A)

(9) A=(Vo)2(Sin(theta))2/2g

The Maximum Apogee (Amax)

(10) Amax=(Vo)2/2g

  • Angle for maximum apogee is 90°
  • Time from launch to impact is just twice the apogee time
  • ti=2ta
  • At impact, the height is zero

The Impact Time (ti)

(11) ti=(2(Vo))(Sin(theta))/g

The Maximum Impact Time (tiMax)

(12) timax=2Vo/g

The ti to ta Ratio

(13) 2=ti / tA

The Range (R)

(14a) R=(2)(Vo)2(Sin(theta))(Cos(theta))/g

(14b) R=(Vo)2(Sin2(theta))/g

The Maximum Range (Rmax)

(15) Rmax=(Vo)2/g

The Range to Apogee Ratio (R:A)

(16) R/A =4/tan(theta)

The Rmax to Amax Ratio (Rmax:Amax)

(17) Rmax/Amax = 2

  • Every projectile launcher on every planet is capable of shooting twice as far as high

The Shape of the Flight Path (y=f(x))

  • This is the equation that relates the height to the horizontal distance of a projectile
  • It is a very important and useful equation

(18) y(x)=xtan(theta) – (g)(x2)/(2)(Vo)2(Cos(theta))2

The “Artillery Equation”

  • Determines the required launch angle to hit a ground target at a given distance

(19)(theta) = 1/2 sin-1 (gR/(Vo)2)

“Through the Hoop”

  • This equation determines the required launch angle to hit an airborne target at a given distance

(20) (theta)=tan-1(Vo2 +- Vo4 – g2 x2 – 2gyVo2/gx)

The Direction TO the Projectile

  • This is the equation that determines the angle that a laser, located at the projectile’s launch site, must be aimed to hit the projectile in flight

(21) theta(t)=tan-1(tan(theta) – (g)(t) / (2)(Vo)(Cos(theta)))

The Direction OF the Projectile

  • This is the equation that determines the angle that the projectile is traveling with respect to the horizontal

(22) a(t)=tan-1(tan – (g)(t)/(Vo)(Cos(theta))

The Distance to the Projectile

(23) r(t) = t/2 (4(Vo2)-4(Vo)(t)(g)(sin)+g2t2)1/2

The Speed of the Projectile

  • The speed at which the projectile is traveling at any point along its flight path

(24) v(t) = ((Vo2) – (2)(Vo)(t)(g)(sin) + (g2) (t2))1/2

Linear Forces

Force

  • A force is an interaction of fermions with a boson field

Fermion

  • Can be Elementary (not made of smaller) particles
  • Can be Composite (made up of smaller) particles
  • Have mass; therefore move at v < c
  • Have half-integer spin
  • Include electrons, neutrinos, quarks

Bosons

  • Can be Elementary (not made of smaller) particles
  • Can be Composite (made up of smaller) particles
  • Do not have mass; therefore move at v = c
  • Have integer spin
  • Include photons, W & Z, Higgs (aka “The God Particle”).

Field

  • A region of energy surrounding a fermion and composed of bosons
  • The type of bosons in the field determine the type of force that results (e.g., gravity, electric/magnetic, etc.).

The Four Forces of Nature

  • Nature has been shwon to have four forces

  • In order from strongest to weakest

  1. Strong Nuclear

  2. Electric/Magnetic

  3. Weak Nuclear

  4. Gravity