Two-Dimensional Kinematics

Kinematics is a division of Dynamics, 
which focuses on Moving Objects...


In a 2-dimensional motion (Projectile Motion), 
there 2 basic components:




1.) Horizontal Motion
a_x = 0 ;  // horizontal acceleration is set to zero;
v_x = v_0x ;  // horizontal speed is set as initial horizontal speed;
x = v_0x  .  t ;  // horizontal distance is the product of  initial horizontal speed and time in seconds;


2.) Vertical Motion
a_y = -g ;  // vertical acceleration is equal to negative value of the gravitational force;
v_y = v_0y - (g . t) ; // vertical speed;
y = v_0y  .  t   - (1/2 g . t²) ;  // vertical distance;




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Basic Equations:


g   =  9.8 m/s² ; // Gravitational Force;

v_y  =  (√2g . h ) ; // Vertical Speed;


v₀  =  R  (√ g / 2h ) ; // Initial Speed;


R   =  v₀   (√2h / g ) ; // Range or Horizontal Distance;


h   =  ( 1/2 g . t² ) ;  // Height or Vertical Distance;


 t   =  (√2h / g ) ; // Time or Duration;

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Displacement:

R   =  Δ x = x - x₀ = v_0x  .  t =  (v_0x  .  t ) +  (1/2 a_x  . t² ) ;  // Range;

h    =  Δ y = y - y₀  = v_0y  .  t   - (1/2 g . t²) =  (v_0y  .  t ) +  (1/2 a_y  . t² ) ;  // Height;

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Velocity-Initial:

V_0x  = ( V₀ . cos  θ );

V_0y  = ( V₀ . sin  θ )  - ( g . t  );


Velocity-Final:

V_x  = ( V₀ . cos  θ₀ ) =  V_0x + a_x . t    ; 

V_y  = ( V₀ . sin  θ₀ ) =  V_0y + a_y . t    ; 


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Velocity of Projectile:

And after solving the x and y components, you must get the actual value for velocity (speed)
by using Pythagorean Theorem:     v² =  v_x² ₊ v_y²;


V =   √ Vx^2 + Vy^2  ;  // Formula in solving for : velocity (speed) of a projectile;

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Angle of Projectile:

tan θ = (v_y / v_x ) ; 

θ = tan^-1 (v_y / v_x ) ; // Formula in solving for : Angle of Inclination; 

Note : Cotangent is  : cot  or  tan^{-1   ...}

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Max or Peak:

y_peak   =   ( V₀ . sin  θ₀ )  . t_peak   - ( 1/2 g . ( t_peak )² )  ;

 Δ x  =   ( 2 V₀² . [sin  θ₀ ]  . [cos  θ₀ ] ) / ( g ) ;

Δ y_peak  =  ( 2 V₀² . [sin  θ₀ ] ² ) / ( 2 g ) ;

t_peak  =  ( V₀ . sin  θ₀ ) / ( g ) ;


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Time:

t = 2(t_peak) = ( 2 V₀ . sin  θ₀ ) / (g) ;


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Basic Formulas in PHYSICS

In relation with Motion and the 3 Newton's Laws ... in the field of KINEMATICS:


The Standard Values:

V =  Velocity / Speed;
d  =  Displacement / Distance;
t = Time / Duration;
a = Acceleration / Change in Speed / Change in Velocity;


Formulas for V:

V = a . t ;    

V = d / t ;



Formulas for d:

d = V. t ;    //linear distance formula: used when "No Change in Speed" or "Acceleration".

d = 1/2 a . t² ;    //quadratic distance formula: used when finding the Displacement of the Projectile on its highest peak before it goes down, when there is "Change in Speed" or "Acceleration".



Formulas for t:

t = V / a ;  

t = d / V ; 


Formulas for a:

a = V / t ;

a = V² / 2 d ;

a = 2 d / t² ;


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Fundamental Formulas:

MASS = (density) x (volume per unit) ;
FORCE = (mass) x (acceleration) ;
WORK = (force) x (displacement);
POWER  = (force) x (velocity);
MOMENTUM  = (mass) x (velocity);
WEIGHT  = (mass) x (gravitational force);


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Law of Conservation of Energy:

"Energy can never be created nor be destroyed... Energy can change."

Ei = Ef ;  // initial energy = final energy ;

PEi + KEi = PEf + KEf ; // sum of initial PE & KE = sum of final PE & KE ;

m.g.Yi +  1/2 m.Vi² = m.g.Yf + 1/2 m.Vf² ;


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One-Dimensional Motion (Free-Falling Motion 
with Constant Acceleration):





Y = height / vertical distance in meters;
(X = range / horizontal distance);


Vi = initial speed in meters/seconds;
Vf = final speed in meters/seconds;

m = mass in kilograms;

g = gravitational force = acceleration inclined with the gravity of Planet Earth (constant value for gravitational force = 9.8 m/s²);

Solving for ΔY:

m.g.Yi + 1/2 m.Vi² = m.g.Yf + 1/2 m.Vf² ; // This the Initial Equation.

m.g.Yf - m.g.Yi  = 1/2 m.Vi² - 1/2 m.Vf² ; // Step 1 : Arrange ;

m.g.Yf - m.g.Yi  = 1/2 m.Vi² - 1/2 m.Vf² ; // Step 2 : Eliminate common terms ;

[g.Yf - g.Yi  = 1/2 Vi² - 1/2 Vf² ] 1/2 g  ; // Step 3 : Multiply each side by its common terms ;

Yf - Yi  = (Vi² - Vf² ) / (2 g) ; // This is the Result or Final Equation ;


Note: 'ΔY' is equal to 'final Y' minus 'initial Y'.