Motion in Two Dimensions

Projectiles

Projectiles

• Under the influence of gravity alone, near the Earth’s surface an object accelerates downward at 9.8 m/s2.

• This acceleration is vertical and does not affect the horizontal velocity.

3.3 Projectile Motion

Example 3 A Falling Care Package

The airplane is moving horizontally with a constant velocity of +115 m/s at an altitude of 1050m. Determine the time requiredfor the care package to hit the ground.

y ay vy voy t-1050 m -9.80 m/s2 0 m/s ?

3 . 3 P r o j e c t i l e M o t i o n

?0 m / s- 9 . 8 0 m / s 2- 1 0 5 0 m

tv o yv ya yy

221 tatvy yoy 2

21 tay y

s 6.14

sm9.80

m 1050222

ya

yt

Example: The Velocity of the Care Package

What are the magnitude and direction of the final velocity ofthe care package?

3.3 Projectile Motion

14.6 s0 m/s?-9.80 m/s2-1050 m

tvoyvyayy

y ay vy voy t-1050 m -9.80 m/s2 ? 0 m/s 14.6 s

sm143

s 6.14sm80.90 2

tavv yoyy

3.3 Projectile Motion

Conceptual Example 5 I Shot a Bullet into the Air...

Suppose you are driving a convertible with the top down.The car is moving to the right at constant velocity. You pointa rifle straight up into the air and fire it. In the absence of airresistance, where would the bullet land – behind you, aheadof you, or in the barrel of the rifle?

3.3 Projectile Motion

Example 6 The Height of a Kickoff

A placekicker kicks a football at and angle of 40.0 degrees andthe initial speed of the ball is 22 m/s. Ignoring air resistance, determine the maximum height that the ball attains.

3 . 3 P r o j e c t i l e M o t i o n

ov

oxv

oyv

sm1440sinsm22sin ooy vv

sm1740cossm22sin oox vv

3.3 Projectile Motion

14 m/s0-9.80 m/s2?

tvoyvyayy

3 . 3 P r o j e c t i l e M o t i o n

1 4 m / s0- 9 . 8 0 m / s 2?

tv o yv ya yy

yavv yoyy 222 y

oyy

a

vvy

2

22

m 10

sm8.92

sm1402

2

y

3.3 Projectile Motion

Example 7 The Time of Flight of a Kickoff

What is the time of flight between kickoff and landing?

3.3 Projectile Motion

?14 m/s-9.80 m/s20

tvoyvyayy

3 . 3 P r o j e c t i l e M o t i o n

?1 4 m / s- 9 . 8 0 m / s 20

tv o yv ya yy

221 tatvy yoy

2221 sm80.9sm140 tt

t2sm80.9sm1420

s 9.2t

3 . 3 P r o j e c t i l e M o t i o n

E x a m p l e 8 T h e R a n g e o f a K i c k o f f

C a l c u l a t e t h e r a n g e R o f t h e p r o j e c t i l e .

m 49s 9.2sm17

221

tvtatvx oxxox

3.3 Projectile Motion

Conceptual Example 10 Two Ways to Throw a Stone

From the top of a cliff, a person throws two stones. The stoneshave identical initial speeds, but stone 1 is thrown downwardat some angle above the horizontal and stone 2 is thrown atthe same angle below the horizontal. Neglecting air resistance,which stone, if either, strikes the water with greater velocity?