1. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 1 Unit 4 Two Dimensional Kinematics Projectile Motion

2. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 2 Projectile Motion  Projectile motion is the branch of kinematics that deals with motion in two physical dimensions (x and y).  The link between these two physical dimensions is the temporal dimension referred to as time.  In this unit we will study two types of projectile motion.  The first type of problem we will call the Full Range of Motion (FRM) problems.  The second problem type will be called Table Problems (TP).  We will look at the problem types in the following slides.  Before looking at these problem types, we must make a change to our Know Want Table.

3. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 3 The Modified Know Want Table  When we do projectile motion problems, we must make an alteration to the Know Want table.  This change takes into account the two dimensional nature of projectile motion problems.  The kinematics equations will also be modified.  Remember, the link between the two dimensions is the time. HorizontalVertical VKWVKW ag v 1x v 1y v 2x v 2y x1x1 y1y1 x2x2 y2y2 tt VKW a v1v1 v2v2 d1d1 d2d2 t

4. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 4 Table Problems  The below animation provides an example of a TP problem.  In these problems a projectile is launched horizontally from above ground level with only an initial horizontal velocity (the initial vertical velocity is always equal to zero).  Afterwards, the projectile falls to ground level.  The vertical acceleration is always -9.8 m/s 2.  The horizontal acceleration is always 0.0 m/s 2 (the horizontal speed is constant).  The graph of a TP problem appears as follows.

5. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 5 Table Problems and Time  The following experiment demonstrates the relationship between time and the x and y dimensions.  Watch the animation and observe the time it takes for the yellow ball and the red ball to fall.  What did you notice about the times for these two balls?  What type of projectile motion is represented by the yellow ball?  What do you observe about the time it takes for the yellow ball to fall in comparison to the time it took for it to travel its horizontal distance?  The times are the same.  In table problems the time of fall is the same as the time of horizontal travel.

6. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 6 Projectile Motion Example  A fighter jet traveling at an altitude of 500 m at a speed of 112 m/s drops a bomb on MeanyBot.  What type of a projectile motion problem is this one?  Some common questions you will be asked about TP problems are as follows.  What are the initial vertical acceleration and speed of the bomb?  What are the initial horizontal acceleration and speed of the bomb?  How many seconds before being directly overhead should the pilot drop the bomb?  What is the horizontal distance between the plane and MeanyBot when the bomb is dropped?  This problem is similar to WS 13 number 9.

7. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 7 WS 13 Number 9  An airplane traveling 1001 m above the ocean at 125 km/h is to drop a box of supplies to shipwrecked victims below.  How many seconds before being directly overhead should the pilot drop the box?  What is the horizontal distance between the plane and the victims when the box is dropped? HorizontalVertical VKWVKW ag v 1x v 1y v 2x v 2y x1x1 y1y1 x2x2 y2y2 tt

8. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 8 Full Range of motion problems  The below animation provides an example of a FRM problem.  In these problems a projectile is launched from ground level with an initial vertical and horizontal velocity.  Afterwards, the projectile returns to ground level.  The vertical acceleration is always -9.8 m/s 2.  The horizontal acceleration is always 0.0 m/s 2 (the horizontal speed is constant).  The graph of a FRM problem appears as follows.

9. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 9 Projectile Motion Example  An archer fires an arrow at a target as shown.  What type of a projectile motion problem is this one?  Some common questions you will be asked about FRM problems are as follows.  How long will the arrow be in flight?  What will be the maximum height of the arrow.?  How far will the arrow travel horizontally?  In FRM problems you will need to find the “half time” of the projectile’s flight.  Remember, time is the link between the physical dimensions x and y.  This problem is from WS 13 number 5.  Before we do this problem, we will need to discuss the concept of half time and the velocity triangle.

10. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 10 Half Time  The most frequent questions asked for FRM problems are what is the maximum height reached by the projectile and how far did it travel horizontally (range).  In order to answer these questions, you must take into account some basic facts of flight.  For instance, what is the vertical speed of the projectile at its highest point (refer to WS 12)?  v y equals zero at the highest point.  Suppose the potatoes took 4.0 s to go up and back down.  At what time did they reach the highest point?  They reached the highest point at half time (2.0 s).  How much of the total horizontal distance did the potato fired at an angle travel in 2.0 s?  Half of the horizontal distance.  To find the maximum height of the projectile you must find the half time.  To find the range you double the half time. tt 1/2

11. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 11 Setting the Standard  When we do problems involving kinematics, it is important that we stick to a standard when imputing data into the know-want table.  This standard enables us to take into account the vector nature of acceleration, velocity, displacement, etc.  These standards are especially important in multiple body kinematics problems.  The slightest error with a negative will result in numerous miscalculations and be very frustrating.  The following examples will illustrate this point.

12. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 12 The Velocity Triangle  In order to do FRM problems, we must find the initial horizontal and vertical velocities of the projectile.  In order to fid these velocities we will use the velocity triangle.  Problem 5 from WS 13 states that an arrow is fired with a velocity (v) of 49.0 m/s at an angle (  ) of 30  with the horizontal.  This velocity has a horizontal component and a vertical component.  These components can be found by using trigonometry. +x +y

13. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 13 WS 13 Number 5  An Arrow is fired with a velocity of 49.0 m/s at an angle of 30  with the horizontal.  How high will the arrow go?  How far will the arrow travel horizontally? HorizontalVertical VKWVKW ag v 1x v 1y v 2x v 2y x1x1 y1y1 x2x2 y2y2 tt +x +y

14. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 14  A fighter jet traveling at an altitude of 500 m at a speed of 112 m/s drops a bomb on MeanyBot.  What are the initial vertical acceleration and speed of the bomb?  What are the initial horizontal acceleration and speed of the bomb?  How many seconds before being directly overhead should the pilot drop the bomb?  What is the horizontal distance between the plane and MeanyBot when the bomb is dropped?  This problem is similar to WS 13 number 9. Projectile Motion Example (WS 10 3)

15. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 15 The Jet Problem  A fighter jet traveling at an altitude of 500 m at a speed of 112 m/s drops a bomb on MeanyBot.  What are the initial vertical acceleration and speed of the bomb?  What are the initial horizontal acceleration and speed of the bomb?  How many seconds before being directly overhead should the pilot drop the bomb?  What is the horizontal distance between the plane and MeanyBot when the bomb is dropped?  This problem is similar to WS 13 number 9. HorizontalVertical VKWVKW ag v 1x v 1y v 2x v 2y x1x1 y1y1 x2x2 y2y2 tt

16. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 16 WS 14 Number 2  An angry physics teacher is launched from a cannon by his students with a velocity of 18.8 m/s at an angle of 65 .  How high above the ground did the bald-headed grouch fly?  How far will he fly horizontally before landing? HorizontalVertical VKWVKW ag v 1x v 1y v 2x v 2y x1x1 y1y1 x2x2 y2y2 tt

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