2What we have been studying: So far all of the motion we have studied as been in a straight line. This is also called linear motion. Not everything always moves in a straight line. A football that is thrown will move in both the vertical and horizontal directions at the same time.
3Vectors and ScalarsA vector is a quantity that has magnitude and direction. A scalar is a quantity that has only magnitude. Velocity is a vector that we have already been using. Two other vectors that we have studied are displacement and acceleration.
4Vectors All vectors have magnitude and direction 7 m/s west magnitude direction
5Vectors Vectors are represented as arrows. Every vector has a tail and a head.
6VectorsThe length of the vector indicates the magnitude
7Vector Addition Vectors can be added together. The resultant is the sum of vectors.
9Vector AdditionVectors can even be added if they are not in the same or opposite directions.We can add vectors if they are at right angles.To add vectors at right angles we will use the Pythagorean theorem.
13Vector AdditionWhen vectors are at right angles to one another the magnitude of the resultant is determined with the Pythagorean theorem. The direction is a combination of the two contributing vectors.
16What is a projectile?A projectile is an object that moves in two dimensions at the same time.One of the dimensions must be vertical.A projectile is acted on only by gravity in the vertical direction.An example of a projectile is any thrown object.
18Why does a projectile move the way it does? A projectile moves only under the influence of gravity.This is not to say that there was no force that got it moving horizontally but as soon as the projectile is thrown or fired or released, there is no longer a horizontal force present.
20Horizontal MotionWhen we neglect air resistance (which we will always do for problem solving), a projectile moves at a constant speed in the horizontal direction.The horizontal acceleration of a projectile is zero.
21Vertical MotionA projectile is influenced by gravity in the vertical direction.This means an object will slow down on the way up and speed up on the way down just like an object that was thrown straight up would do.The vertical acceleration of a projectile is 10 m/s2.
24Horizontal and Vertical Motion The horizontal motion and the vertical motion are independent of one another.This means that gravity is going to work the same whether a ball is thrown softly, thrown very fast or even just dropped.
25Check for Understanding A ball is dropped off a cliff at the same time a cannon ball is fired horizontally off the same cliff. Which will hit the ground first?
27Check for Understanding A ball is dropped off a cliff at the same time a cannon ball is fired horizontally off the same cliff. Which will hit the ground first?They will both hit the ground at the same time. Gravity does the same thing to both of them.
29Check for Understanding Suppose a rescue airplane drops a relief package while it is moving with a constant horizontal speed at an elevated height. Assuming that air resistance is negligible, where will the relief package land relative to the plane?a. below the plane and behind it.b. directly below the planec. below the plane and ahead of it
30Check for Understanding Suppose a zookeeper must shoot a banana from a banana cannon to a monkey who hangs from the limb of a tree. This particular monkey has a habit of dropping from the tree the moment that the banana leaves the muzzle of the cannon. The zookeeper is faced with the dilemma of where to aim the banana cannon in order to hit the monkey. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon?
33Fast Moving Projectiles SatellitesA satellite is any object which is orbiting the earth, sun or other massive body.Satellites can be categorized as natural satellites or man-made satellites.
34Fast Moving Projectiles The moon, the planets and comets are examples of natural satellites. Accompanying the orbit of natural satellites are a host of satellites launched from earth for purposes of communication, scientific research, weather forecasting, intelligence, etc
35Fast Moving Projectiles A satellite is a projectile.That is to say, a satellite is an object upon which the only force is gravity. Once launched into orbit, the only force governing the motion of a satellite is the force of gravity. Newton was the first to theorize that a projectile launched with sufficient speed would actually orbit the earth.
38Projectile Motion Sample Problem Frank stands on the balcony of a building and kicks a pebble off with a horizontal velocity of 5 m/s. The balcony is 12 m high. How long will it take the pebble to reach the ground and how far horizontally will it travel?
39Sample ProblemFirst pick out the given numbers in the problem and organize them as either vertical or horizontal.Next write which equation matches which informationHorizontal Vertical v = 5 m/s Dy = 12 m a = 10 m/s2
40Sample Problem s = Dx/t Dy = ½ at2 Horizontal Vertical v = 5 m/s Dy = 12 ma = 10 m/s2
41Sample ProblemWhy have I changed the equations from s = d/t and d = ½ at2 To s = x/t and y = ½ gt2 ??????????????????????????????????????????
42Sample ProblemNow find time with either the vertical or horizontal information. NEVER NEVER EVER mix horizontal and vertical information into the same equation!
43Sample ProblemWHICH IS BEST TO FIND TIME? s = Dx/t Dy = ½ at2 Horizontal Vertical v = 5 m/s Dy = 12 m a= 10 m/s2 THE ONE WITH THE MOST INFORMATION.
44Sample Problem Dy = ½ at2 Vertical y = 12 m g = 10 m/s2 Dy = ½ at2 12 = ½ (10) t2 12 = 5 t = t2 t = 1.55 sDy = ½ at2Verticaly = 12 mg = 10 m/s2
45Sample ProblemHave we answered the problem? Frank stands on the balcony of a building and kicks a pebble off with a horizontal velocity of 5 m/s. The balcony is 12 m high. How long will it take the pebble to reach the ground and how far horizontally will it travel? Not quite through.
46Sample Problems = Dx/t Horizontal v = 5 m/s Now we also know time; t = 1.55 s 5 = x/ * 1.55 = x Dx = 7.75 m
47Now you tryA plane flying at 70 m/s drops a package from a height of 5000 m. How long will it take the package to reach the ground and how far horizontally will it travel?