Kinematics in Two Dimensions; Vectors Chapter 3 Kinematics in Two Dimensions; Vectors

Units of Chapter 3 Vectors and Scalars Addition of Vectors – Graphical Methods Subtraction of Vectors, and Multiplication of a Vector by a Scalar Adding Vectors by Components Projectile Motion Solving Problems Involving Projectile Motion Projectile Motion Is Parabolic (qualitative) Relative Velocity

3-1 Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature

Vector operations you need to know How to resolve (“break”) a vector into x and y components practice How to find resultant vector starting with x and y components How to add vectors by adding x and y components

3-2 Addition of Vectors – Graphical Methods For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates.

3-2 Addition of Vectors – Graphical Methods If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

3-2 Addition of Vectors – Graphical Methods Adding the vectors in the opposite order gives the same result:

3-2 Addition of Vectors – Graphical Methods Even if the vectors are not at right angles, they can be added graphically by using the “tail-to-tip” method. http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

3-2 Addition of Vectors – Graphical Methods The parallelogram method may also be used; here again the vectors must be “tail-to-tip.”

3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction. Then we add the negative vector:

3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar A vector V can be multiplied by a scalar c; the result is a vector cV that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

22 sept: 3-4 Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.

3-4 Adding Vectors by Components If the components are perpendicular, they can be found using trigonometric functions.

3-4 Adding Vectors by Components The components are effectively one-dimensional, so they can be added arithmetically:

3-4 Adding Vectors by Components Draw a diagram; add the vectors graphically. Choose x and y axes. Resolve each vector into x and y components. Calculate each component using sines and cosines. Add the components in each direction to give Resultant. To find the length and direction of the resultant vector, use:

Vector operations you need to know How to resolve (“break”) a vector into x and y components practice How to find resultant vector starting with x and y components How to add vectors by sketching How to add vectors by adding x and y components Practice

Vector check Look at this velocity vector, V. What is Vx? Include units What is Vy? Include units

Vector motion Your pen is a rocket Move your rocket according to the vectors drawn on the board. Do not stab your neighbor

3 Oct: 3-5 Projectile Motion A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola. Anybody know the equation of a parabola?

Predict the range Shoot projectile straight up, measure H Shoot projectile horizontal off table, predict R Shoot projectile at 45o, predict R Shoot projectile at 60o, predict R

3-5 Projectile Motion It can be understood by analyzing the horizontal and vertical motions separately. They each happen at the same time but independently of each other

3-5 Projectile Motion The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

3-5 Projectile Motion If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component. This is why we learned vectors!

Another view of projectile motion Vertical shot from a moving cart Show the cart with a shot Show the loggerpro ball shot vertical from cart\ball shot vertical from cart.cmbl Answer the questions.

3-6 Solving Problems Involving Projectile Motion Projectile motion is motion with constant acceleration in each of two dimensions, in x-direction: a = 0 And in the y-direction, a = -g Start with general kinematics and show these.

“The Big 10” Problem solving steps Draw and label the situation at two times; define in each of x and y. Always draw the parabola. Resolve any vector used into x and y components Label knowns and unknowns (t,x,y,vx,vy,g) You must choose the time interval for the problem This time interval must have only freefall involved Choose one dimension to start Identify the correct kinematic equation for that x or y Usually have to solve for t to link x and y Solve algebra for desired unknown Find equation and solve for other dimension x or y Check the units of the algebraic equations Do the math with units Check answer for sig figs

Practice, practice, practice The classics you will learn Car -- horizontal off a cliff Cannon projectile -- on the level Ball up off a cliff, find final velocity only Ball down off a cliff, find final velocity only Throw a ball up to a cliff ledge

3-7 Projectile Motion Is Parabolic (option) In order to demonstrate that projectile motion is parabolic, we can use x= and y= equations with substitution to see that: This is indeed the equation for a parabola.

The WWI German “Paris Gun” The gun was capable of shooting a 94 kg (210 lb) shell to a range of 130 km (81 mi) and a maximum altitude of 40 km (25 miles, 131,000 ft) — the greatest height reached by a human-made projectile. At the start of its 170s trajectory, each shell from the Paris Gun reached a speed of 1,600 m/s. With these numbers, is the projectile behaving according to our assumptions?

10 Oct: Classic problems chapter 3 Cannon on the level Car horizontal off a cliff Ball up off a cliff, find final velocity only Ball down off a cliff, find final velocity Throw a ball up to a cliff ledge

12 oct: Labs in science classes Learning about science vs. Doing science Hypothesis: a well-defined proposition that can be reproducibly tested Experiment: a reproducible process to test a given hypothesis “lab” = experiment Prelab report: the preplanning for a lab (see physics tools) Lab report: summary of the lab (see physics tools) Look at their notes, this is not in the text

3-8 Relative Velocity We already considered relative speed in one dimension; it is similar in two dimensions except that we must add and subtract velocities as vectors. Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity. Therefore, vWS is the velocity of the water in the shore frame, vBS is the velocity of the boat in the shore frame, and vBW is the velocity of the boat in the water frame.

3-8 Total Velocity To find total velocity, simply add the two velocities that are acting together = vboat + vriver This is adding velocity vectors just like adding displacement vectors to find overall displacement

Relative velocity “Relative” means one compared to the other. 16.2 m/s What is velocity of the hobo relative to the ground? 16.2 m/s What is velocity of the dog relative to the cat? vcat It is vdog - vcat vdog

Summary of Chapter 3 A quantity with magnitude and direction is a vector. A quantity with magnitude but no direction is a scalar. Vector addition can be done either graphically or using components. The sum is called the resultant vector. Projectile motion is the motion of an object near the Earth’s surface under the influence of gravity.

practice

Some questions for review Review symmetry of vdown and vup for vertical motion What happens to tf or vf if h doubles? If throw up off a cliff and down off a cliff at same vyo, why does object reach the ground with same vy?

Quiz After reviewing ball off a cliff. I drop the ball with the GoMotion on the ceiling. Falls in about 0.6s distance of 2.8m. Derive formula for time to fall Calculate numerical result for time Compare to observed time

High jump video problem https://youtu.be/qB0Fq3UXEqA is a nice analysis of high jump.  he jumps 93 inches, but his center of mass we'll assume goes up to 87 inches.  Assume he lands back on the ground and not on a high pad.  How long would he be in the air?  I estimate that his forward travel, the Range, is 72 inches.   So what is his vx at takeoff?  what is the angle θ of his launch?  Height H = 87” = 2.20m range R = 72” = 1.82m Solve t setting time to top of curve: t = (2H/g)1/2 = 0.671s So total time = 1.34s Also can solve by finding vyo first vyo = (2gH)1/2 = 6.58m/s vxo = 2.72m/s from R and t value And can find  from tan-1 = 6.58/2.72 = 67o

Practice projectile problems HW Build classic list

Quiz on projectile lab In the lab, there is one variable (condition varied to see the effect of the change). What is the variable, or what is varied? What is the measured quantity in the lab; what could change as the variable changes? Sketch the projectile path that you will use in the lab.

Lab 2: projectile motion of a ball Use handout Sketch the problem on the handout Equipment and video analysis Will you be here on Thursday for the full period? Come to class Thursday ready to do the lab. Groups: In work sessions.

Final quiz/problem Easier harder I throw the timer-ball at known angle, you catch at time t. You know distance. What is vo? How high did the ball go? harder Set up in class, and solve for next class I throw this ball and it goes R distance in t time. What is vo and q of the launch?

Quiz for Felix Buamgartner jump -- 14 Oct 2012 Started at 128,100 ft above earth = 39.0km Reached ~729 mph after ~48 seconds = 326m/s Calculate average a. 6.79m/s2 Would you expect a to be constant? Why? Is it higher or lower than g? Why? How far does he fall in 48 s? Can’t know since a isn’t constant.