Lecture 6 Purdue University, Physics 220 1

Lecture 06

Projectile Motion

Textbook Sections 4.2

PHYSICS 220

Lecture 6 Purdue University, Physics 220 2

Relative Velocity

• We often assume that our reference frame is attached to the Earth. What

happen when the reference frame is moving at a constant velocity with

respect to the Earth?

• The motion can be explained by including the relative velocity of the

reference frame in the description of the motion.

The ground velocity of an

airplane is the vector sum of

the air velocity and the wind

velocity. Using the air as the

intermediate reference frame,

ground speed is:

Example airplane

V(PG)=V(PA) +V(AG)

Lecture 6 Purdue University, Physics 220 3

Three swimmers can swim equally fast relative to the water. Theyhave a race to see who can swim across a river in the least time.Relative to the water, Beth (B) swims perpendicular to the flow, Ann(A) swims upstream, and Carly (C) swims downstream. Whichswimmer wins the race?

A) Ann

B) Beth

C) Carly

correct

A B C

x

y

Exercise

!

t = d / vy

Ann vy = v cos(!)

Beth vy = v

Carly vy = v cos(!)Lecture 6 Purdue University, Physics 220 4

What angle should Ann take to get directly to the other

side if she can swim 5 mph relative to the water, and the

river is flowing at 3 mph?

A B C

VAnn,ground = Vann,water+Vwater,ground

x

y

x-direction:

sin(!) = |Vwater,ground|/ |Vann,water|

sin(!) = 3/5

!

Exercise

Lecture 6 Purdue University, Physics 220 5

y

x

2-Dimensions

• X and Y are INDEPENDENT!

No component of one F or v on other

They don’t have to be vertical and horizontal just

at right angles to each other

• Break 2-D problem into two

1-D problems

Demo 1D-21 was done in L04

Lecture 6 Purdue University, Physics 220 6

Velocity in Two Dimensions

5 m/s

3 m/s

A ball is rolling on a horizontal surface at 5 m/s. It then rolls

up a ramp at a 25 degree angle. After 0.5 seconds, the ball

has slowed to 3 m/s.

What is the magnitude of the change in velocity?

y

x

x-direction

vix = 5 m/s

vfx = 3 m/s cos(25)

"vx = 3cos(25)–5 =-2.28m/s

y-direction

viy = 0 m/s

vfy = 3 m/s sin(25)

"vy = 3sin(25)=+1.27 m/s

!v = !v

x

2+ !v

y

2

= 2.6 m/s

Lecture 6 Purdue University, Physics 220 7

Acceleration in Two DimensionsA ball is rolling on a horizontal surface at 5 m/s. It then rolls

up a ramp at a 25 degree angle. After 0.5 seconds, the ball

has slowed to 3 m/s.

What is the average acceleration?

5 m/s

3 m/s = 5.21 m/s

2

a = a

x

2+ a

y

2

y

x

x-direction y-direction

ax=!2.28m/s

0.5 s = !4.56 m/s

2

a

y=

1.27m/s

0.5 s = 2.54 m/s

2

Lecture 6 Purdue University, Physics 220 8

Kinematics in Two Dimensions

x and y motions are independent!

They share a common time t

x = x0 + v0xt + 1/2 axt2

vx = v0x + axt

vx2 = v0x

2 + 2ax "x

y = y0 + v0yt + 1/2 ayt2

vy = v0y + ayt

vy2 = v0y

2 + 2ay "y

Lecture 6 Purdue University, Physics 220 9

Projectile Motion

x = x0 + v0x t

vx = v0x

y = y0 + v0y t - ! gt2

vy = v0y – g t

vy2 = v0y

2 – 2 g "y

x-direction: ax = 0 y-direction: ay = -g

Lecture 6 Purdue University, Physics 220 10

Velocity of a Projectile

Velocity components of a projectile

Independence of the Vertical and

Horizontal motion of Projectiles

Lecture 6 Purdue University, Physics 220 11 Lecture 6 Purdue University, Physics 220 12

A place-kicker kicks a football at an angle of !=400

above the horizontal axis. The initial speed of the

ball is v0=22 m/s. Ignore air resistance and find the

range R that the ball attains.

v0=22m/s

!=400

H

R

The Range of a Kickoff

Lecture 6 Purdue University, Physics 220 13

y = y0 + v0yt - 1/2 gt2

vy = v0y - gt

vy2 = v0y

2 - 2g "y

x = x0 + v0xt

vx = v0x

• Use the equations

v0x = v0cos ! =

(22m/s)cos 400 = 17 m/s v0x!

v0yv0

The Range of a Kickoff

• The range is a characteristic of the horizontal motion

• You need v0x and v0y but you have been given v0

Lecture 6 Purdue University, Physics 220 14

The Range of a Kickoff

v0x!

v0yv0

v0y =v0sin !=

(22m/s)sin 400=14 m/s

• We could be done if we know the time of flight of the kickoff

• The time of flight can be determined from y equations. For

example the time to get to height H is

• Therefore the time to determine the range is #2.9 s

x = R = v

0xt

vy= v

0 y! gt

th=

v0 y! v

y

g=

14m / s

9.8m / s2= 1.428s

x = R = v

0xt = (17m / s)(2.9s) = 49m

• The range depends on the angle ! at which the football is kicked.

Maximum range is reached for !=450

Lecture 6 Purdue University, Physics 220 15

Range of a Projectile Demo 1D-22

•Two ways to hit a target except at limit

Lecture 6 Purdue University, Physics 220 17

You are a vet trying to shoot a tranquilizer dart into a monkey hangingfrom a branch in a distant tree. You know that the monkey is verynervous, and will let go of the branch and start to fall as soon as your gungoes off. On the other hand, you also know that the dart will not travel ina straight line, but rather in a parabolic path like any other projectile. Inorder to hit the monkey with the dart, where should you point the gunbefore shooting?

A) Right at the monkey

B) Below the monkey

C) Above the monkey

Shooting the Monkey

•Demo 1D-23

Lecture 6 Purdue University, Physics 220 18

yy = vv0y t - 1/2 gg t2

yy = y0 - 1/2 gg t2

Dart hits the

monkey!

Shooting the Monkey