HERE IS AN EXAMPLE OF ONE-DIMENSIONAL MOTION YOU’VE SEEN BEFORE SO FAR IN THIS COURSE YOU HAVE BEEN STUDYING WHAT IS CALLED ONE-DIMENSIONAL MOTION ONE-DIMENSIONAL MOTION HAS A STRAIGHT LINE PATH – OBJECTS MOVE BACK AND FORTH ALONG THIS ONE STRAIGHT LINE HERE IS AN EXAMPLE OF ONE-DIMENSIONAL MOTION YOU’VE SEEN BEFORE

IMAGINE THE RED ARROW REPRESENTS THE MAN’S DISPLACEMENT DRAWN ON A SET OF AXES HIS MOTION IS PARALLEL TO ONLY ONE AXIS THAT IS WHY IT IS ONE DIMENSIONAL MOTION

HERE’S ANOTHER EXAMPLE AGAIN A SET OF AXES WITH A DISPLACEMENT VECTOR ARROW IS DRAWN IN THE MOTION OF THE BALL IS PARALLEL TO ONLY ONE AXIS (THE VERTICAL AXIS THIS TIME) SO THE BALL’S MOTION IS ONE DIMENSIONAL

BUT SUPPOSE AN OBJECT’S MOTION IS PARALLEL TO MORE THAN ONE AXIS, LIKE THIS GIRL’S MOTION AGAIN AXES ARE DRAWN AND HER DISPLACEMENT VECTOR IS DRAWN FROM WHERE SHE STARTS TO WHERE SHE STOPS, CONNECTING THE ENDS OF HER CURVED PATH SHE STARTS HERE PART OF HER MOTION IS PARALLEL TO THE HORIZONTAL AXIS AND ANOTHER PART IS PARALLEL TO THE VERTICAL AXIS – THUS HER MOTION IS PARALLEL TO TWO AXES THIS IS TWO-DIMENSIONAL MOTION SHE STOPS HERE

THE FIRST EXAMPLE OF TWO-DIMENSIONAL MOTION WE WILL STUDY WILL BE PROJECTILE MOTION A PROJECTILE IS DEFINED AS: ANY OBJECT THAT IS LAUNCHED INTO FREE FALL NO MATTER WHAT THE OBJECT IS, IF IS LAUNCHED THEN ALLOWED TO FREE FALL, THAT OBJECT IS CALLED A PROJECTILE AND IS IN WHAT PHYSICISTS CALL PROJECTILE MOTION

THE PROJECTILES WE WILL STUDY NOW WILL BE IN TWO-DIMENSIONAL MOTION THE DIVER SHOWN AT THE RIGHT IS MOVING HORIZONTALLY TO THE LEFT AT THE SAME TIME HE IS MOVING VERTICALLY DOWNWARD THIS IS TWO-DIMENSIONAL MOTION AND IT PRESENTS US WITH A PROBLEM WE NEED TO RESOLVE

THEY CANNOT BE USED WHENEVER YOU PLEASE. HERE ARE THE THREE KINEMATIC EQUATIONS WE HAVE USED THESE EQUATIONS ARE RESTRICTED THEY CANNOT BE USED WHENEVER YOU PLEASE. WHAT ARE THE RESTRICTIONS ON THE KINEMATIC EQUATIONS? THE ACCELERATION (a) MUST BE CONSTANT IN BOTH DIRECTION AND MAGNITUDE 2. THE OBJECT MUST BE IN ONE-DIMENSIONAL MOTION THIS LAST RESTRICTION IS THE PROBLEM MENTIONED EARLIER: THE KINEMATIC EQUATIONS ARE GOOD ONLY FOR ONE-DIMENSIONAL MOTION, BUT WE WANT TO STUDY TWO-DIMENSIONAL PROJECTILE MOTION WHAT’S A PHYSICIST TO DO??

HERE’S HOW WE WILL RESOLVE THIS PROBLEM. THE DIVER IS FALLING AND MOVING TO THE LEFT SIMULTANEOUSLY. TWO POINTS IN HIS MOTION HAVE BEEN INDICATED BY RED DOTS WE WILL CALL THE TOP DOT THE DIVER’S ORIGINAL POSITION, AND THE BOTTOM DOT HIS FINAL POSITION THE GREEN ARROW INDICATES HIS DISPLACEMENT VECTOR ALONG THE VERTICAL AXIS THE ORANGE ARROW INDICATES HIS DISPLACEMENT VECTOR ALONG THE HORIZONTAL AXIS

HERE’S HOW WE WILL RESOLVE THIS PROBLEM. THE GREEN LINE INDICATES HIS DISPLACEMENT ALONG THE VERTICAL AXIS THE DIVER’S VERTICAL MOTION BY ITSELF IS ONE DIMENSIONAL SO WE CAN IGNORE HIS HORIZONTAL MOTION, AND CONSIDER HIS VERTICAL MOTION ALONE. BECAUSE WE ARE CONSIDERING ONLY THE VERTICAL ONE-DIMENSIONAL MOTION, WE ARE ALLOWED TO USE THE KINEMATIC EQUATIONS.

HERE’S HOW WE WILL RESOLVE THIS PROBLEM. THE ORANGE ARROW INDICATES HIS DISPLACEMENT VECTOR ALONG THE HORIZONTAL AXIS USING THE SAME LOGIC: THE DIVER’S HORIZONTAL MOTION BY ITSELF IS ONE DIMENSIONAL SO WE CAN IGNORE HIS VERTICAL MOTION, AND CONSIDER HIS HORIZONTAL MOTION ALONE. BECAUSE WE ARE CONSIDERING ONLY THE HORIZONTAL ONE- DIMENSIONAL MOTION, WE ARE ALLOWED TO USE THE KINEMATIC EQUATIONS.

HERE’S HOW WE WILL RESOLVE THIS PROBLEM. PROBLEM SOLVED! WE CAN USE THE KINEMATIC EQUATIONS IF WE USE THEM ON THE HORIZONTAL AND VERTICAL MOTIONS SEPARATELY INSTEAD OF LOOKING AT PROJECTILE MOTION AS ONE TWO-DIMENSIONAL MOTION, WE WILL LOOK AT IT AS TWO SEPARATE ONE-DIMENSIONAL MOTIONS.

IN ORDER TO EXECUTE THIS SEPARATION OF MOTIONS ACCURATELY, YOU WILL NEED TO FOLLOW A SERIES OF STEPS IN EVERY PROBLEM. THESE STEPS ARE CALLED THE VECTOR COMMANDMENTS VECTOR COMMANDMENTS READ THE PROBLEM CAREFULLY, THEN DRAW A SKETCH OF THE SITUATION DESCRIBED, INCLUDING THE SIGN CONVENTION YOU WILL USE. THE DRAWING MUST INCLUDE ALL NECESSARY VECTORS AS ARROWS. a) The sketch must show all given quantities. b) The “necessary vectors” are those vectors that appear in the equations you will use c) If the problem has a drawing in it, use that drawing instead of making your own, but make sure all necessary vectors and given quantities are shown 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS

SINCE WE ARE GOING TO DEAL WITH THE VERTICAL AND HORIZONTAL MOTIONS SEPARATELY, WE NEED TO INDICATE WHICH OF THE TWO WE ARE WORKING WITH. WE WILL DO THE WITH THE SUBSCRIPTS (x) AND (y). THESE ARE THE KINEMATIC EQUATIONS IN THEIR ORIGINAL FORM HERE ARE THE KINEMATIC EQUATIONS FOR HORIZONTAL MOTION WHY DOESN’T THE (t) HAVE AN (x) SUBSCRIPT? HERE ARE THE KINEMATIC EQUATIONS FOR VERTICAL MOTION ON A PIECE OF SCRATCH PAPER, WRITE OUT THE KINEMATIC EQUATIONS FOR VERTICAL MOTION.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? READ THE PROBLEM CAREFULLY, THEN DRAW A SKETCH OF THE SITUATION DESCRIBED, INCLUDING THE SIGN CONVENTION YOU WILL USE. WHEN MAKING THE SKETCH YOU INCLUDE ONLY THOSE DETAILS WHICH ARE ESSENTIAL TO THE PROBLEM, AND YOU DO NOT ATTEMPT TO MAKE A BEAUTIFUL ARTIST’S RENDERING OF THE SITUATION THE SKETCH AT THE RIGHT SHOWS THE SIGN CONVENTION, THE CLIFF AND THE TIGER (REPRESENTED BY THE RECTANGLE). THESE ARE THE ESSENTIALS.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? READ THE PROBLEM CAREFULLY, THEN DRAW A SKETCH OF THE SITUATION DESCRIBED, INCLUDING THE SIGN CONVENTION YOU WILL USE. THE DRAWING MUST INCLUDE ALL NECESSARY VECTORS AS ARROWS. a) The sketch must show all given quantities. b) The “necessary vectors” are those vectors that appear in the equations you will use HERE IS THE SKETCH WITH THE NECESSARY VECTORS AND THE GIVEN QUANTITIES INCLUDED.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol ONE OF THE NECESSARY VECTORS WAS THE ACCELERATION VECTOR (a). HOWEVER, WE CANNOT USE ANY OF THE KINEMATIC EQUATIONS IF THE ACCELERATION IN NOT CONSTANT. WHAT DO WE KNOW ABOUT THE ACCELERATION OF ANY PROJECTILE, SUCH AS THE TIGRESS, WHICH IS BY DEFINITION AN OBJECT IN FREE FALL?

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol WRITE OUT THE i-j NOTATION FOR THE TIGRESS’ ACCELERATION AS SHE FLIES THROUGH THE AIR. IS THIS WHAT YOU WROTE?

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol BASED ON THE INFORMATION IN THE PROBLEM AND THE SKETCH, WRITE OUT THE (VO) IN i-j NOTATION IS THIS WHAT YOU WROTE?

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol THE LAST VECTOR WE NEED TO WRITE OUT IS THE DISPLACEMENT VECTOR (S). THE VERTICAL COMPONENT WAS GIVEN IN THE PROBLEM AND IS SHOWN IN THE SKETCH HOWEVER, WE DO NOT KNOW A NUMERICAL VALUE FOR THE HORIZONTAL COMPONENT. WRITE OUT THE i-j NOTATION FOR THE (S) VECTOR

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol YOU MIGHT THINK THIS IS CORRECT BUT AS IT SAYS ABOVE “Any quantity for which you do not have a value must be written using the appropriate mathematical symbol” WHAT IS THE CORRECT SYMBOL FOR THE HORIZONTAL COMPONENT OF THE DISPLACEMENT?

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 2. WRITE OUT ALL NECESSARY VECTORS IN i-j NOTATION a) All necessary vectors must be in i-j notation – even if you do not have all the numerical values. Any quantity for which you do not have a value must be written using the appropriate mathematical symbol THE SYMBOL FOR DISPLACEMENT IS (S) THE SYMBOL FOR HORIZONTAL IS THE SUBSCRIPT (x). THEREFORE, THE CORRECT SYMBOL FOR HORIZONTAL DISPLACEMENT IS ( SX ) AND THE i-j NOTATION FOR THE DISPLACEMENT VECTOR IS

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? THE SKETCH AND EQUATIONS AT THE RIGHT SHOW THE RESULTS OF THE FIRST TWO STEPS OF THE VECTOR COMMANDMENTS. ALL OF THIS IS ONLY PREPARATION FOR FINDING THE ANSWER, WE HAVEN’T YET BEGUN TO WORK OUT THE SOLUTION, WHICH IS VECTOR COMMANDMENT STEP 3.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS ANSWER THE FOLLOWING QUESTIONS ON SCRATCH PAPER WHAT IS IT WE ARE ASKED TO FIND? WRITE OUT THE KINEMATIC EQUATION FOR FINDING THE VALUE OF (Sx ). WE ARE ASKED TO FIND THE TIGRESS’ HORIZONTAL DISPLACEMENT, THAT IS, THE (Sx )

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS ANSWER THE FOLLOWING QUESTIONS ON SCRATCH PAPER IN ORDER TO FIND (SX ), FOR WHAT QUANTITIES DO WE NEED VALUES? WE NEED (vox ), (ax ), and (t) DO WE HAVE VALUES FOR ALL THESE QUANTITIES? NO – WE DO NOT HAVE A VALUE FOR (t), THE TIME THE TIGRESS WAS FLYING THROUGH THE AIR.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS IN ORDER TO USE THE EQUATION ABOVE, WE MUST FIRST GET A VALUE FOR THE TIME (t) NOTE THAT THE TIME (t) THE TIGRESS TOOK TO FLY OUTWARD HORIZONTALLY WAS THE SAME AMOUNT OF TIME IT TOOK HER TO FREE FALL THE (3.8 M) FROM THE TOP OF THE CLIFF IN OTHER WORDS, THE HORIZONTAL TIME (t) IS THE SAME AS THE VERTICAL TIME, ALSO (t).

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS WE CAN USE THE VERTICAL MOTION TO FIND OUT THE HORIZONTAL (AND VERTICAL) TIME(t) ANSWER THE FOLLOWING QUESTION ON SCRATCH PAPER WRITE OUT THE EQUATION FOR CALCULATING THE VERTICAL DISPLACEMENT IS THIS WHAT YOU WROTE?

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS ANSWER THE FOLLOWING QUESTION ON SCRATCH PAPER FROM THE EQUATION ABOVE AND THE VALUES GIVEN BELOW THE SKETCH AT THE RIGHT, WHAT IS THE TIME THE TIGRESS IS FLYING THROUGH THE AIR, BOTH VERTICALLY AND HORIZONTALLY?

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS NOTICE THAT WE GET TWO ANSWERS FOR THE TIME (t). HOWEVER, SINCE TIME CAN NEVER BE NEGATIVE WE THROW (-0.88) OUT BECAUSE IT IS PHYSICALLY IMPOSSIBLE.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS ANSWER THE FOLLOWING QUESTION ON SCRATCH PAPER NOW THAT WE KNOW THE TIGRESS WAS IN THE AIR FOR (0.88) SECONDS, GO AHEAD AND CALCULATE HOW FAR HORIZONTALLY SHE FLEW.

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? 3. AFTER YOU HAVE COMPLETED STEPS 1 AND 2, YOU CAN BEGIN TO SOLVE THE PROBLEM USING SELECTED KINEMATIC EQUATIONS HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? THE TIGRESS LANDS 21.21 TO THE RIGHT OF THE BASE OF THE CLIFF

EXAMPLE PROBLEM A TIGRESS RUNS ALONG A LOW CLIFF, WHICH IS (3.8 m) HIGH, AT A SPEED OF (24 m/s). SHE LEAPS HORIZONTALLY FROM THE CLIFF AND LANDS SAFELY ON THE GROUND. HOW FAR FROM THE BASE OF THE CLIFF DOES SHE LAND? THIS IS THE ENTIRE SOLUTION TO THE PROBLEM ABOVE

TWO IMPORTANT POINTS 1. YOU MUST FOLLOW THE VECTOR COMMANDMENTS, IN THE PROPER ORDER, LEAVING NOTHING OUT 2. SOLVING A PROJECTILE MOTION PROBLEM OFTEN REQUIRES THE USE OF MORE THAN ONE KINEMATIC EQUATION.