© 2001-2005 shannon w. helzer. all rights reserved. 1 unit 4 two dimensional kinematics projectile...

© 2001-2005 Shannon W. Helzer. All Rights Reserved. 1

Unit 4Two Dimensional Kinematics

Projectile Motion


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.


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.

Horizontal Vertical

V K W V K W

a g

v1x v1y

v2x v2y

x1 y1

x2 y2

t t

V K W

a

v1

v2

d1

d2

t

atvv 12

122

12

2 2 ddavv

2112 2

1attvdd

2 1x x xv v a t

2 22 1 2 12x x xv v a x x

22 1 1

1

2x xx x v t a t

2 1y yv v gt

2 22 1 2 12y yv v g y y

22 1 1

1

2yy y v t gt


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/s2. The horizontal acceleration is always 0.0 m/s2 (the horizontal speed is constant). The graph of a TP problem appears as follows.

x v. y

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35

x (m)

y (

m)


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.


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.


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?

Horizontal Vertical

V K W V K W

a g

v1x v1y

v2x v2y

x1 y1

x2 y2

t t


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/s2. The horizontal acceleration is always 0.0 m/s2 (the horizontal speed is

constant). The graph of a FRM problem appears as follows.

x v. y

0

25

50

75

100

125

0 5 10

x (m)

y (

m)


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.


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)?

vy 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.t t1/2


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.


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

v

yv

xv

sinopp

hyp

cosadj

hyp

yv

v

xv

v

sinv

cosv


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?

Horizontal Vertical

V K W V K W

a g

v1x v1y

v2x v2y

x1 y1

x2 y2

t t+x

+y

v

yv

xv



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)


The Jet Problem


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.

Horizontal Vertical

V K W V K W

a g

v1x v1y

v2x v2y

x1 y1

x2 y2

t t


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?

Horizontal Vertical

V K W V K W

a g

v1x v1y

v2x v2y

x1 y1

x2 y2

t t

v

yv

xv


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© 2001-2005 shannon w. helzer. all rights reserved. 1 unit 4 two dimensional kinematics projectile...

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