If you know your projectile motion, you can make sledding a bit more fun

Or you can perform feats like.???

Demo: Shoot the Monkey!Where should the banana cannon be aimed in order for the monkey to catch the banana, IF the monkey lets go of the branch at the instant the banana is fired?Aim Above, At, or Below??Aim ______ the Monkey!AT

Why?? Lets take a look1st a Review of Free Fall:How far does an object fall if g = 10 m/s2 in 1 second? _____ 2 seconds? _____ 3 seconds? _____Either of the following formulas can be used:d = _____d = _____5 m20 m45 m

With those free-fall distances in mind, lets take another look at a horizontal projectilePath if NO gravity5 m20 m45 m

Again, with those free-fall distances in mind, lets now take a look at a projectile that was launched at an angle5 m20 m45 m

So, how does this explain the Shoot the Monkey Demo?Monkey here initially Monkey here after 3 seconds of free fall the same place as the banana at this time!!

How does the initial velocity of the banana (projectile) affect this situation?The higher the velocity, the _______ the banana is caught by the monkey. Higher Initial VelocityLower Initial Velocityhigher

What would happen if the banana cannon were aimed above the monkey?

An Example Problem:A football leaves a punters foot at 25 m/s and at a 60 angle with the horizontal. Find the horizontal velocity and the initial vertical velocity.Vi = 25 m/s60So, if the punter were moving horizontally at ______ and then threw the football directly up at _______, the resulting trajectory of the football would be exactly the same as if he had punted it at 25 m/s at a 60 angle.Vx Viy =12.5 m/s=21.7 m/s12.5 m/s21.7 m/s

Since the force of gravity is down, it can only decelerate the ___________ component of the initial velocity. verticalOr use the velocity formula:Calculate the hang time (total time in the air) for the football, assuming it was punted and caught at the same height above the ground.

Recall that the acceleration in the x direction is ______, so the distance is ONLY dependent on ________________ and ________________.Calculate the horizontal distance that the football traveled. ZEROhorizontal speedtotal time

Calculate the footballs maximum height above where it was kicked. The time to reach the max height is ______, which in this case is ______.Defining the up direction as +

A look at the Velocity Vectors: another animation from The Physics Classroom website...

10 m/s10 m/s10 m/s10 m/s10 m/s10 m/s10 m/s10 m/sHow would the speed at 1 sec compare to the speed at 7 sec?010 m/s2 (down)40 m/s30 m/s20 m/s10 m/s10 m/s20 m/s30 m/s40 m/sIf tup = 4 s, viy = 10 m/s2 x 4s

Same Speed! v2=102+302After adding vectors head to tailA look at the velocity values for a projectile launched at an angle:

A look at a projectile thrown down at an angle:5 m20 m45 m5 m20 m45 mt = 1 s - t = 2 s - t = 3s - 3.5 s

A Review Example:Joe Physics, accident investigator extraordinaire, is asked to determine whether or not a car was speeding before it crashed through the guard rail of a bridge as shown. The posted Speed Limit was 55 mph (24 m/s). What is his conclusion? Hint: the car is a ___________ projectile.horizontalThe time to reach the ground below is ____________________________.the same time as to free fallBusted! Why?

Question: What angle should a projectile be fired to produce the maximum range (horizontal distance when projectile begins and ends at the same height)?To answer this question, we are going to find an expression for the horizontal distance (range) traveled given the initial velocity vi and the angle q.

vi qFind expressions for vx & viy:vx viy Find an expression for the time in the air:

Find an expression for the horizontal distance (range):At this point, we can go no further in our derivation until we discover a trig identity

What are Trig Identities?An equation involving trig functions that are true for all possible values of the variable. For example:

Try any angle q and the equation is true!

A double-angle trig identity:

Back to our equation for the Range:Substituting the double-angle trig identity:This equation is very useful when you are asked for the range or horizontal distance of a projectile, but you do NOT know the time!

Use this Range Equation to investigate what launch angle will produce the greatest range:If vi = 20 m/s & q =

Conclusions:A _____ launch angle will produce the maximum range!sin(2q) = sin___ = ___ in this case.

_____________ angles (angles that add up to _____) produce the same range! *But, this is true ONLY if no air resistance. With air, a launch angle a little ______ than 45 will produce the maximum range! Complementaryless

The following website can be found by googling Projectile Motion Java Applet:

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

Investigate different angles with and without air.