Physics 1-2 Mr. Chumbley
Physical quantities can be categorized into one of two types of quantities A scalar is a physical quantity that has magnitude, but no direction A vector is a physical quantity that has both magnitude and direction
We have already been using symbols to represent physical quantities Vector and scalar quantities are identified by how they are written Scalars are written using italics Vectors can be written two ways In boldface: x, v, a, F An arrow above the symbol: x, v, a, F
Vectors are drawn using lines with arrowpoints The length of the vector is representative of the magnitude has a greater magnitude than The direction of the arrow indicates the direction
When drawing vectors, it is important to know that vectors can be moved to any location, so long as the magnitude and direction remain the same This is especially useful when combining multiple vectors
In order for vectors to be added together, they must have he same units and describe the same quantities Vectors can be added both mathematically and graphically The resultant is a vector that represents the sum of two or more vectors
One of the simplest ways to add vectors is by drawing them graphically Tail Tip Moving all vectors so that the tip of one vector connects to the tail of the next vector The resultant vector can be drawn from the tail of the first vector to the tip of the last vector
Determining the vectors graphically can be done in different ways The magnitude and direction of the vector can be measured using a ruler and protractor The magnitude and direction can be calculated if the vector is placed in a scaled grid
For vectors, there are a couple different ways in which direction can be indicated One way is to treat the direction as if the vector were rotated from one of the cardinal directions The second method is to treat the direction as if the vector were rotated from a standard direction of 0, similar to a coordinate plane
As stated before, the magnitude of the vector is represented by the length of the arrow To accurately depict a vector, it is often necessary to use a scale to represent those quantities
When looking at a vector in two dimensions, the vector can be broken down into two onedimensional vectors v The one-dimensional part of a twodimensional vector is known as a component v y For our purposes vectors will generally have a y-component and an x-component v x
If the magnitude and direction of a vector are known, determining the components is as simple as making a right triangle v y can be found: v y = v sin(θ) v x can be found: v x = v cos(θ) θ v x v v y
An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid s height is 136 m and its width is 2.30 10 2 m. What is the magnitude and direction of the displacement of the archaeologist after she has climbed from the bottom to the top? Given Δy = 136 m Δx = ½ (width) = 115 m Unknown d =? θ =? d θ 115 m 136 m
Find the components of the velocity of a helicopter travelling 95 km/h while rising at an angle of 35 to the ground Given V = 95 km/h θ = 35 Unknown v y =? v x =? 35 v y v x
P. 87 #1-4 P. 90 #1-4
Adding vectors that are not perpendicular can be complicated Using the standard trig functions does not work since there is not a right triangle However, by breaking each vector into components, a right triangle can be created to solve for the resultant sum
Given: d 1 = 27.0 km θ 1 = -35 A hiker walks 27.0 km from her base camp at 35 south of east. The next day, she walks 41.0 km in a direction 65 north of east and discovers a ranger s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower. d 2 = 41.0 km θ 2 = 65 Unknown: d =? θ =? Diagram: d θ d 2 θ 1 Δy d 1 θ 1 2 Δx 2 Δy 2
P. 92 #1-4
Projectile motion is the curved path than an object follows when thrown, launched, or otherwise projected near the surface of Earth
When describing the motion of an object in two dimensions, it can be done in two ways The first is by using the vector quantities of displacement, velocity, and acceleration This would result in vector multiplication The second is by looking at the twodimensional motion in its two onedimensional components This avoids vector multiplication entirely
When a projectile is broken down into its component parts, it is revealed that the vertical and horizontal motion occur independently This means that the projectile is simultaneously undergoing free fall, and moving with constant horizontal velocity This motion results in the projectile following a parabolic trajectory
Projectiles can be treated as an object in free fall with an initial horizontal velocity Since the only acceleration is in the vertical direction, the horizontal velocity is constant Quantity Vertical Horizontal Displacement y = v yi t 1 2 g( t)2 x = v x t Velocity v yf = v yi g t v yf 2 = v yi 2 2g y v xf = v xi = constant Acceleration a y = g = 9.81 m/s 2 a x = 0 m/s 2
The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horizontally off the bridge. The magnitude of the rock s horizontal displacement is 45.0 m. Find the magnitude of the velocity at which the rock was kicked. Given: Δy = -321 m Δx = 45.0 m v yi = 0 m/s a y = -g = -9.81 m/s 2 a y v x Unknown: v x =? -321 m 45.0 m
P. 97 #1-4
When an object is launched at an angle, it then has both initial vertical and horizontal velocities Therefore, finding the components of the initial velocity can be done through vectors v xi = v i cos θ v yi = v i sin θ
The equations that describe the motion of the projectile do not change Horizontal velocity remains constant Vertical motion still behaves as free fall
A shell is fired from the ground with an initial speed of 1.70 10 3 m/s at an initial angle of 55.0 to the horizontal. a. How much time is the shell in the air? b. How far from where it is launched does the shell land? Given: v i = 1.70 10 3 m/s a y = -g = -9.81 m/s 2 Unknown: t =? x =? v i x
P. 99 Practice #1-4
Looking at the path of a projectile, the velocity at any point can be determined One of the more beneficial points to determine the velocity is at the end of the projectile s motion The final velocity is the vector sum of the horizontal velocity (v x ) and the final vertical velocity (v yf )
The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horizontally off the bridge at a speed of 5.56 m/s. The magnitude of the rock s horizontal displacement is 45.0 m. What is the velocity of the rock just before it hits the ground below? Given: Δy = -321 m Δx = 45.0 m v yi = 0 m/s v x = 5.56 m/s a y = -g = -9.81 m/s 2 a y v x -321 m Unknown: v f =? 45.0 m θ f v x v yf v f
Calculate the final velocity of the objects from problems #1-4 on p. 97 Quantity Vertical Horizontal Displacement y = 1 2 a y( t) 2 x = v x t Velocity v yf = a y t v yf 2 = 2a y y v xf = v xi = constant Acceleration a y = g = 9.81 m/s 2 a x = 0 m/s 2
A boat heading north crosses a wide river with a velocity of 10.00 km/h relative to the water. The river has a uniform velocity of 5.00 km/h due east. Determine the boat s velocity with respect to an observer on the shore. Given: v boat = 10.00 km/h v river = 5.00 km/h Unknown: v observed =? θ =?
P. 103 #1-4