1. Projectile Motion and Centripetal Force

2. Projectile Motion
Projectile motion: motion that travels along a curved path.
This curve is a combination of horizontal motion (motion along the x-axis) and vertical motion (motion along the y-axis).
Ex: A ball thrown thru the air, a cannonball launched

3. Projectile: any object that moves through the air or space, acted on only by gravity (and air resistance, if any)
Accelerating in the vertical direction due to the force of gravity
Moving at a constant velocity in the horizontal direction due to inertia
Ex: sports ball, object in free fall, bullet

4. The horizontal motion (motion along the x-axis) of a projectile is at a constant velocity because there is NO force in the horizontal direction.
It keeps moving in the x-axis direction because of inertia!
The vertical motion of a projectile is at constant acceleration because gravity is ALWAYS acting on it.
Gravity slows the upward motion, and gravity speeds up the downward motion of the projectile.

6. THE HORIZONTAL MOTION FOR A PROJECTILE IS COMPLETELY INDEPENDENT OF THE VERTICLE COMPONENT OF MOTION!!!!!!!!!!!!
The combined effects is what produces the curved paths that projectile follow.

7. Check Your Understanding
At the instant a horizontally pointed cannon is fired, a cannonball held at the cannon’s side is released and drops to the ground. Which cannonball strikes the ground first, the one fired from the cannon or the one dropped?
Both cannonballs fall the same vertical distance with the same acceleration due to gravity; therefore they strike the ground at the same time. (Remember that horizontal velocity is independent of vertical velocity).

8. Range of Projectiles Horizontal range Vertical range
Projectiles will have a maximum horizontal range if launched at a 45 degree angle
As the angle moves away from 45 degrees, the horizontal range decreases
89 and 1 degrees will have the shorter ranges
44 and 46 will have longer ranges
Vertical range
The steeper the angle at which a projectile is launched, the higher the path will be.
Projectiles launched at steep angles do not travel far in the horizontal direction.
When air resistance is significant, the range of the projectile is below its ideal.

9. Check Your Understanding
A projectile is launched into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?
Its vertical acceleration is g (10 m/s2) because the force of gravity is downward. Its horizontal acceleration is zero because no horizontal force acts on it.

10. Check Your Understanding
At what point in its path does a projectile have minimum speed?
The minimum speed of a projectile occurs at the top of its path. The speed at the top is the horizontal velocity as there will be no vertical velocity at the peak.

11. Rotation & Revolution
Axis: a straight line around which rotation takes place
Objects spin on their axis
Ex: the North-South pole line thru the earth
Rotation: axis is located within the body of the object
An object spinning about on its axis
Ex: spinning skater

12. Revolution: when an object turns about an external axis
Object moves AROUND another object
Ex: earth revolves around the sun in 365 days

13. Rotation vs. Revolution
The turntable rotates about its axis.
The lady bug revolves around the same axis

14. Check Your Understanding
Does a tossed football rotate or revolve?
rotates about its own axis
Does a ball whirled overhead at the end of a string rotate or revolve?
it revolves around you

15. Rotational Speed vs Linear Speed
Linear Speed: distance moved per unit of time
varies with the distance moved from the axis
the further away from the axis, the greater the linear speed
Rotational Speed: number of rotations per unit time
does NOT vary with distance from the axis

16. Rotational Speed vs. Linear Speed
All parts of the turntable rotate at the same speed, but the ladybugs at different distances from the center travel at different linear speeds.
A ladybug sitting twice as far from the center moves twice as fast.

17. Check Your Understanding
Which part of Earth’s surface has the greatest rotational speed relative to Earth’s axis?
all parts of Earth have the same rotational speed.
Which part of Earth’s surface has the greatest linear speed relative to Earth’s surface?
The equator has the greatest linear speed because it is furthest from the axis.

18. Check Your Understanding
If a meter stick supported at the 0cm mark swings like a pendulum from your fingers, how fast at any given moment is the 100cm mark moving compared to the 50cm mark?
Twice as fast for linear speed because the 100 cm mark is twice as far from the axis of rotation. The rotational speed is the same everywhere.

19. Centripetal Force
Centripetal force: any force that causes an object to follow a circular path
Centripetal means center seeking or towards the center
SI Unit: Newtons (N)
Equation: Fc = mv2 / r
Fc = centripetal force (N)
m = mass (kg)
v = velocity (m/s)
r = radius (m)

20. Mass and centripetal force are directly proportional
2m = 2FC
½m = ½FC
10m = 10FC
1/10m = 1/10FC
Radius and centripetal force are inversely proportional
2r = ½FC
½r = 2FC
10r = 1/10FC
1/10r = 10FC

21. As an object moves faster around a curve, the velocity is directly squared to the centripetal force
2v = 22 or 4F
3v = 32 or 9F
10v = 102 or 100F
**This is why you must decrease your speed considerably when going around a curve in your car**

22. Examples of Centripetal Forces
As a car makes a turn, the force of friction acting upon the turned wheels of the car provides the centripetal force required to keep the car in circular motion.

23. As a bucket of water is tied to a string and spun in a circle, the force of tension acting upon the bucket provides the centripetal force required for circular motion

24. As the moon orbits Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

25. Check Your Understanding
A motorcycle runs on the inside of a bowl-shaped track. Is the force that holds the motorcycle in a circular path an inward- or outward- directed force?
It is an inward-directed force – also known as a centripetal force.

26. When the string breaks, the whirling can moves in a straight line, tangent to its circular path. Notice that the can does not move outward from the center

27. !!!!Important !!!!
In order for circular motion to take place, a centripetal force must be present!
The reason that objects are “pushed” to the outside of a circle is because they have inertia!
Ex: your clothes being pushed to the outside of the washing machine during the spin cycle; your behind remaining in the seat on a loop-de-loop on a roller coaster.

28. Check Your Understanding
A tin can, with a mass of 1 kg, is on a string with a length of 2 m. If the can is being whirled around someone’s head at 4 m/s, what is the centripetal force acting on the can?
Fc = ?
m = 1 kg
v = 4 m/s
r = 2 m
Fc = mv2 / r
Fc = (1)(4)(4) / (2) = 8 Newtons

29. Check Your Understanding
If the string on a tin can suddenly breaks as it is being whirled overhead, which way does it fly?
In a straight line tangent to the circle.
Why?
It’s inertia will allow it to move in the direction that it was already moving in (in non-accelerated motion…not turning or changing speed).