Kinematics Where? When How fast? Free fall Projectile Motion

The equations of kinematics Constant speed x = v t Constant acceleration v = v 0 + a t starting from rest x = ½ a t 2 Some examples (see the board)

Examples 1.An object starts from rest and accelerates uniformly at 10 m/s 2. Find all kinds of stuff. 2.An object starts from rest and after moving for 6 seconds, its speed is 30 m/s. How far has it travelled in those 6 seconds?

More examples 3.An object is initially moving at 3 m/s. It accelerates for 5 seconds, at which time it is moving at 20 m/s. What was its acceleration? 4.Can we find the distance travelled during those 5 seconds?

Free fall: a special case All objects, independent of mass, and ignoring air resistance fall towards the center of the earth with an acceleration “g” g = 9.8 m/s 2 = 32 ft/s 2 How fast is this? Human vs. Gravity

Useful “conversions” Fastest person in the world runs approximately 10 m/s = 22 mph. (???) Watch your units!! 60 mph = 88 ft/s = 27 m/s => Note that “Miles per hours” ≈ 2 × “meters per sec”. Fastest person in the world accelerates at 2 m/s 2.

Free fall, dropping from rest Special case of general kinematics for objects starting from rest v = a × t becomes v = g × t = gt x = ½ × a × t 2 becomes y = ½ × g × t 2 = ½gt 2 Now lots of examples

Free fall from rest examples 1.Object is dropped from rest from a height of 100 meters. How long to hit the ground? How fast is it moving? 1.Object is dropped from rest from a height of 100 feet. How long to hit the ground? How fast is it moving?

More ffall examples 1.Object is dropped from rest from an unknown height. It hits the ground at 40 m/s. Find the initial height. 2.An object is dropped from rest from a height of 50 meters on the planet Arrisonia. It hits the ground in 10 seconds. How fast does it hit the ground?

What if you throw something up? Back to the original definition of acceleration as a change in speed Can “reason” your way to maximum height Can get rather tricky rather fast, unless you want to do lots of algebra.  Let’s try a couple anyway...

Objects moving up 1.An object is thrown straight up at a speed of 60 ft/sec. How long does it move up? How high does it go? How long until it gets back down to its original position? Total time of flight? 2.An object is thrown straight up at a speed of 30 m/s. How high does it go?

Projectile Motion Combination of vertical motion and horizontal motion. Horizontal motion is constant speed. Why? Inertia! Is that an answer? Vertical Motion is constant acceleration. Why? Free fall! Is that an answer?

Some examples 1. An object slides horizontally off of a 2.0 meter high table. It is moving at 5.0 m/s. How far from the bottom of the table does the object hit? 2. An object slides horizontally off of the roof of a 5.0 m tall building, and hits the ground 20 meters from the edge of the building. How fast did the object slide off of the roof?

Ok, now a tough one... An object is thrown at 30 m/s at an angle of 53.1° above the horizontal from the top corner of a 45 meter tall (15 stories) building. (Note change!) a)What is the maximum height reached? b)Where and when does it hit the ground? c)How fast is it moving when it hits the ground? WOW!! How the $*#&% are we going to do this??

Let’s split it into “easy” parts... 1.Split the initial velocity into horizontal and vertical components. (Trig?? OH NO!) 2.Solve the “going up” part. 3.Solve the “coming down” part. 4.Solve the “moving horizontal” part. 5.Put it all together. And now we’re done!!!

Finally... A weird application of Free Fall, as shown by Isaac Newton in his Principia Mathematica. Maybe surprising is a better word than weird, eh? Okay, go home and do practice problems – the ones that I sent, and more that I will send. Have a great weekend!