1. Page 1-3 is Notes

2. Please answer and show work for pages 4-8

Worksheet 3 - Integral Applications to Motion and the Center of Mass Now use your value of M and your equation for dM to evaluate Xem. You know that dM is the amount of mass that has a position X. This means you can add up all of the masses of the all of the super thin strips that make up the triangle of metal: 30cm [Xam [ Xdm X on = object total mass of object M Evaluate this integral to find the location of the center of mass X value. Xcm = 8Worksheet 3 - Integral Applications to Motion and the Center of Mass 3. From what height did he jump, and what was his final velocity with which he hit the ground? Height = V = 4. Also, from what height could he have jumped without his cape to strike the ground with the same speed? without cape, Height =Worksheet 3 - Integral Applications to Motion and the Center of Mass The Daredevil A daredevil tries jumping off the roof of a building with a loose billowing cape of cloth, hoping the cloth will slow him down. He jumps from an unknown height, yo, and lands at y = 0 two seconds later. While he is falling his velocity is given by the function: V(1)=-9.81+212 Note that here it is assumed he jumps at t = 0 and that up is the positive direction. The presence of the 2t- term represents the action of the cape to slow him down. 1. What is the acceleration as a function of time that he experiences while in the air? A(t) = 2. What is his position as a function of time? Use the integral. Also use the fact that he is in the air for exactly 2 seconds and lands at y = 0 to determine the unknown constant of integration, C. Y(t) = 4PHYS-124 Worksheet 3 - Integral Applications to Motion and the Center of Mass Review - The Anti-Derivative and Understanding the Integral In calculus, a mathematical function called the integral is defined, where the integral is the inverse operator of the derivative. This means that if you take the integral and then the derivative, you will obtain your original function: [ f(x)dx (notation for the derivative of f (x)) J f(x)dx = f(x) This means that when you apply the integral to a function, f(x), you get the function whose derivative is f(x). Hence the name "anti-derivative!" Here are some basic anti-derivatives, or integrals: 1. The sum of two functions: [ [ () + 8(x)] =[ fondx+[=(x)dx 2. Polynomials: [x" dx = 1x"+ + C where C is an arbitrary constant of integration 3. Trig functions: [Asin( Bxkdx = - 4 cos(Bx) + C [Acos( Bx)dx = = sin( Bx) + C 4. Definite integrals: When we evaluate the integral from X1 to x2 we get: [ fondx =[ fedex evanated at x2 - [ f( x)dex evauated at x, For example: [x'dx =-x' +cWorksheet 3 - Integral Applications to Motion and the Center of Mass Finding the Center of Mass A piece of sheet metal is cut into a triangle such that the bottom edge lies along the x axis and the top edge follows the equation y = 0.6x. The triangle goes out to x = 30cm. Each square centimeter has a mass of 0.6 grams. Your task is to find the center of mass of this triangle. To help you do this, I have broken the problem into a series of steps. 1. Evaluating the Xem. Consider the vertical strip of metal that is between Strip of X and X + dX. This strip has a width of only dX. What is width dx its height? Use this to find the area, dA, of this strip of metal. Keep your units in cm, so dA will have units of cm. Use this value of dA to calculate the mass of this thin strip, dM (note that since dA is in cm , dM should be in grams). dA and dM should be functions of X and dX. 30cm dA = dM = What is the total mass of the triangle of metal? M =Worksheet 3 - Integral Applications to Motion and the Center of Mass Let's Play Ball! A 0.5 kg baseball is thrown at a speed of 13m/s. You hit it with the bat, and it goes straight back to the pitcher. This means the collision is one dimensional, along the line between you and the pitcher. The force of collision last for 0.04 seconds and is given by the function: F(1) =590N sin(25x) for 0