Ideal Projectile Equations: If the only force is weight, then the x velocity stays constant (ax = 0). The y velocity changes with time and position (y acceleration ay = -g).

(Ideal) P rojec tile Equations

A c c el:

Veloc ity:

Pos ition:

v = v - 2g(y - y )y y0 022

v = v - gty y0

x y

0 -g

A n additional y equation:

x = x + v0 x t y = y + v t - gt0y2

0 12

S I U nits : U S U nits :

g = 9.81 m/s2

g = 32.2 fps2v = v x 0 c os

v = v 0 0y(w here, )s in

v0v0y

s inv 0=

y

xc osv 0=vx

Remember to use the correct g for your units!

A general projectile motion problem involves seven “pieces” of information [ x0, y0, , xL, yL, and tL ] . General P rojec tile P roblem

L aunch L oc ation:

L anding L oc ation and T ime:

L aunch Veloc ity and Angle: (v ) 0 at

v0

x

y

(x , y )0 0

(x , y )L Lat time, tL

x L

yL

at time, t = 0

(x , y )0 0

(x , y ) at t ime tL L L

General P roblem: Given 5 out of 7 of these “piec es” of info.

U se the tw o pos ition equations to solve for the remaining tw o:

Pos ition: x = x + v0 x t y = y + v t - gt0y2

0 12

w here, v = v x 0 c os v = v 0 0y s in

Usually you are givenfive of these and askedto find the remainingtwo, usually applyingthe two positionequations.

A general projectile motion problem involves seven “pieces” of information [ x0, y0, , xL, yL, and tL ] .

C lass A : “L aunc h Info K now n” P ro jec tile P roblems :

For these, you are given c omplete launch data:

(x , y ),0 0 (v ), and of 0 oneat G iven:

F ind: T he tw o of (x , y , t )remaining L L L

(x , y , t )L L L

1

2

3

Given: [ (x , y ), (v , ), and x ] F ind: (y , t )0 0 0 L L L

Given: [ (x , y ), (v , ), and y ] F ind: (x , t )0 0 0 L L L

Given: [ (x , y ), (v , ), and t ] F ind: (x , y )0 0 0 L L L

5 Given: [ (x , y ), (v , ), and s loped ground] F ind: (x , y , t )0 0 0 L L L

4 Given: [ (x , y ), (v , )] F ind: h , t0 0 0 apex apex (at apex )

For problems through , w rite the tw o pos ition equations

y = y + v t - gt0y2

0 12x = x + v0 xt

and so lve for the tw o unk now ns .

1 3

A general projectile motion problem involves seven “pieces” of information [ x0, y0, , xL, yL, and tL ] .

C lass B P ro jec ti le P roblems :

For these, you are given the launch loc ation ( ),the landing loc ation ( ), and of (v , , t ).one 0 L

x , yx , y

0 0

L L

F ind: T he tw o of ( )remaining v , , t0 L

6 Given: [ (x , y ), ( ), and ] F ind: ( , t )0 0 Lx , y vL L 0

(T his is the mos t c ommon, and the eas iest, ofthese c ases . Write the tw o pos ition equations ,

y = y + v t - gt0y2

0 12x = x + v0 xt

and so lve for v and t .0 L

7 Given: [ (x , y ), ( ), and ] F ind: ( )0 0 x , y t v ,L L L 0

9 Given: [ (x , y ), ( ), and ] 0 0 x , y vL L 0

F ind: Tw o s and tw o ’s for this .’ t vL 0

8 Given: [ (x , y ), ( )] 0 0 x , yL L

F ind: M inimum , and c orresponding and .v t0 L