PROJECTILE MOTION

Projectile Examples

• Tennis ball• Golf ball• Football• Softball• Soccer ball• Bullet

• Hockey puck• Basketball• Volleyball• Arrow• Shot put• Javelin

These are all examples of things that are projected, then go off under the

influence of gravity

Not projectiles• Jet plane• Rocket• Car (unless it looses contact with ground)

The key to understandingprojectile motion is to realize that gravity acts vertically

it affects only the vertical part of the motion, not thehorizontal part of the motion

Understanding Projectiles

Demonstration

• We can see that the horizontal and vertical motions are independent

• The red ball falls vertically

• The yellow ball was given a kick to the right.

• They track each other vertically step for step and hit the ground at the same time

In the absence of gravity a bullet would follow a straight line forever.With gravity it FALLS AWAY fromthat straight line!

Projectile Paths

Shoot the Monkey

Sample Problem

A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, she kneels 10.0 m from the light pole,which is 5.00 m high.

The tip of her gun is 1.00 m above the ground. At the same moment that the monkey drops a banana, the zookeeper shoots.

If the dart travels at 50.0 m/s,will the dart hit the monkey, the banana, or neither one?

1 . Select a coordinate system. The positive y-axis points up, and the positive x-axis

points along the ground toward the pole. Because the dart leaves the gun at a height of 1.00 m, the vertical distance is 4.00 m.

Sample Problem

2 . Use the inverse tangent function to find the angle that the initial velocity makes with the x-axis.

1 1 4.00 mtan tan 21.8

10.0 m

y

x

Sample Problem

3 . Choose a kinematic equation to solve for time.

Rearrange the equation for motion along the x-axis to isolate the unknown Dt, which is the time the dart takes to travel the horizontal distance.

x (vi cos )t

t x

vi cos

10.0 m

(50.0 m/s)(cos 21.8 )0.215 s

Sample Problem

4 . Find out how far each object will fall during this time. Use the free-fall kinematic equation in both cases.

For the banana, vi = 0. Thus:

Dyb = ½ay(Dt)2 = ½(–9.81 m/s2)(0.215 s)2 = –0.227 m

Sample Problem

The dart has an initial vertical component of velocity equal to vi sin q, so:

Dyd = (vi sin q)(Dt) + ½ay(Dt)2 Dyd = (50.0 m/s)(sin 21.8)(0.215 s) +½(–9.81 m/s2)

(0.215 s)2

Dyd = 3.99 m – 0.227 m = 3.76 m

Sample Problem

5 . Analyze the results. Find the final height of both the banana and the dart.

ybanana, f = yb,i+ Dyb = 5.00 m + (–0.227 m)

ybanana, f = 4.77 m above the ground

Sample Problem

The dart hits the banana.

The slight difference is due to rounding.

ydart, f = yd,i+ Dyd = 1.00 m + 3.76 m

ydart, f = 4.76 m above the ground

Sample Problem

No gravity is good for kickers

Newton’s First Law of Motion

• “Every object continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state of motion by forces impressed upon it ”

• The tendency of matter to maintain its state of motion is known as INERTIA.

Basketball – without gravity

Hitting the target – aim high, not directly at the target

BULLSEYE!

Path of the Projectile

v

Distance downfield(range)

Hei

ght rising falling

projectile

g

Horizontal velocity

Verticalvelocity

v

Horizontal Motion

0

0 0

0

cos

x x x

x

x

v v a t

a

v v

210 0 2

0 0 0cos

x xx x v t a t

x x v t

Vertical Motion

210 0 2

210 0 0 2

0

0 0

(2 15)

sin (4 22)

(2 11)

sin (4 23)

y y

y y y

y

y y v t a t

y y v t gt

v v a t

v v gt

Projectile motion – key points

1) The projectile has both a vertical and horizontal component of velocity

2) The only force acting on the projectile once it is shot is gravity (neglecting air resistance)

3) At all times the acceleration of the projectile is g = 9.8 m/s2 downward

4) The horizontal velocity of the projectile does not change throughout the path

Key points, continued

5) On the rising portion of the path gravity causes the vertical component of velocity to get smaller and smaller

6) At the very top of the path the vertical component of velocity is ZERO

7) On the falling portion of the path the vertical velocity increases

More key points

8) If the projectile lands at the same elevation as its starting point it will have the same vertical SPEED as it began with

9) The time it takes to get to the top of its path is the same as the time to get from the top back to the ground.

10) The range of the projectile (where it lands) depends on its initial speed and angle of elevation

A 2.00 m tall basketball player wants to make a basket froma distance of 10.0 m. If he shoots the ball at a 450 angle, atwhat initial speed must he throw the ball so that it goes through the hoop without striking the backboard?

yxy0

Sample Problem

210 0 2

210 0 0 2

0

0 0

(2 15)

sin (4 22)

(2 11)

sin (4 23)

y y

y y y

y

y y v t a t

y y v t gt

v v a t

v v gt

210 0 2

0 0 0cos

x xx x v t a t

x x v t

0

0 0

0

cos

x x x

x

x

v v a t

a

v v

Equations to Choose from

Maximum Range

• When an artillery shell is fired the initial speed of the projectile depends on the explosive charge – this cannot be changed

• The only control you have is over the angle of elevation.

• You can control the range (where it lands) by changing the angle of elevation

• To get maximum range set the angle to 45°

Interactive

• http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ProjectileMotion/jarapplet.html

• http://jersey.uoregon.edu/vlab/Cannon/

• Imagine trying to throw a rock aroundthe world.

• If you give it a large horizontal velocity,it will go into orbitaround the earth!

The ultimate projectile: Orbit