Download - Projectile Motion and Centripetal Force. Projectile Motion Projectile motion: motion that travels along a curved path. This curve is a combination of.

Projectile Motion and Centripetal Force

Projectile Motion

Projectile motion:Projectile motion: motion that travels motion that travels along a curved path.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

Projectile:Projectile: any object that moves any object that moves through the air or space, acted on through the air or space, acted on only by gravity (and air resistance, only by gravity (and air resistance, if any)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

• 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.

• 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.

Check Your UnderstandingCheck Your Understanding

At the instant a horizontally pointed At the instant a horizontally pointed cannon is fired, a cannonball held at cannon is fired, a cannonball held at the cannon’s side is released and the cannon’s side is released and drops to the ground. Which cannonball drops to the ground. Which cannonball strikes the ground first, the one fired strikes the ground first, the one fired from the cannon or the one dropped?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).

• Horizontal 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.

Range of Projectiles


A projectile is launched into the air. A projectile is launched into the air. Neglecting air resistance, what is its Neglecting air resistance, what is its vertical acceleration? Its horizontal vertical acceleration? Its horizontal acceleration?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.

At what point in its path does a projectile At what point in its path does a projectile have minimum speed?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.


Rotation & Revolution

Axis:Axis: a straight line around which a straight line around which rotation takes placerotation takes place Objects spin on their axis Ex: the North-South pole line thru the earth

Rotation:Rotation: axis is located within the axis is located within the body of the objectbody of the object An object spinning about on its axis Ex: spinning skater

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

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


Does a tossed football rotate or revolve?Does a tossed football rotate or revolve? rotates about its own axis

Does a ball whirled overhead at the end Does a ball whirled overhead at the end of a string rotate or revolve?of a string rotate or revolve?

it revolves around you

Rotational Speed vs Linear Speed

Linear Speed:Linear Speed: distance moved per distance moved per unit of timeunit of time varies with the distance moved from the axis the further away from the axis, the greater the

linear speed

Rotational Speed:Rotational Speed: number of number of rotations per unit timerotations per unit time does NOT vary with distance from the axis

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.


Which part of Earth’s surface has the Which part of Earth’s surface has the greatest rotational speed relative to greatest rotational speed relative to Earth’s axis?Earth’s axis?

all parts of Earth have the same rotational speed.

Which part of Earth’s surface has the Which part of Earth’s surface has the greatest linear speed relative to Earth’s greatest linear speed relative to Earth’s surface?surface?

The equator has the greatest linear speed because it is furthest from the axis.

If a meter stick supported at the 0cm If a meter stick supported at the 0cm mark swings like a pendulum from your mark swings like a pendulum from your fingers, how fast at any given moment fingers, how fast at any given moment is the 100cm mark moving compared to is the 100cm mark moving compared to the 50cm mark?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.


Centripetal Force

Centripetal force:Centripetal force: any force that any force that causes an object to follow a circular pathcauses 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)

• 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

• 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**

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.

• 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

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


A motorcycle runs on the inside of a A motorcycle runs on the inside of a bowl-shaped track. Is the force that bowl-shaped track. Is the force that holds the motorcycle in a circular path holds the motorcycle in a circular path an inward- or outward- directed force?an inward- or outward- directed force?

It is an inward-directed force – also known as a centripetal force.

• 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

!!!!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.

Check Your UnderstandingCheck Your UnderstandingA 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


If the string on a tin can suddenly breaks If the string on a tin can suddenly breaks as it is being whirled overhead, which as it is being whirled overhead, which way does it fly?way does it fly?

In a straight line tangent to the circle.

Why?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).