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to Introduction Projectile Motion Learning Goal: To understand the basic concepts of projectile motion. Projectile motion may seem rather complex at first. However, by breaking it down into components, you will find that it is really no different than the one-dimensional motions that you have already studied. One of the most often used techniques in physics is to divide two- and three-dimensional quantities into components. For instance, in projectile motion, a particle has some initial velocity . In general, this velocity can point in any direction on the xy plane and can have any magnitude. To make a problem more managable, it is common to break up such a quantity into its x component and its y component . Consider a particle with initial velocity that has magnitude 12.0 and is directed 60.0 above the negative x axis. Part A What is the x component of ? Express your answer in meters per second. ANSWER: = -6.00 Correct Part B What is the y component of ? Express your answer in meters per second. ANSWER: 10.4 Correct Breaking up the velocities into components is particularly useful when the components do not affect each other. Eventually, you will learn about situations in which the components of velocity do affect one another, but for now you will only be looking at problems where they do not. So, if there is acceleration Introduction to Projectile Motion in the x direction but not in the y direction, then the x component of the velocity will change, but the y component of the velocity will not. Part C Look at this applet. The motion diagram for a projectile is displayed, as are the motion diagrams for each component. The x-component motion diagram is what you would get if you shined a spotlight down on the particle as it moved and recorded the motion of its shadow. Similarly, if you shined a spotlight to the left and recorded the particle's shadow, you would get the motion diagram for its y component. How would you describe the two motion diagrams for the components? ANSWER: Both the vertical and horizontal components exhibit motion with constant nonzero acceleration. The vertical component exhibits motion with constant nonzero acceleration, whereas the horizontal component exhibits constant-velocity motion. The vertical component exhibits constant-velocity motion, whereas the horizontal component exhibits motion with constant nonzero acceleration. Both the vertical and horizontal components exhibit motion with constant velocity. Correct As you can see, the two components of the motion obey their own independent kinematic laws. For the vertical component, there is an acceleration downward with magnitude . Thus, you can calculate the vertical position of the particle at any time using the standard kinematic equation . Similarly, there is no acceleration in the horizontal direction, so the horizontal position of the particle is given by the standard kinematic equation . Now, consider this applet. Two balls are simultaneously dropped from a height of 5.0 Part D How long . does it take for the balls to reach the ground? Use 10 for the magnitude of the acceleration due to gravity. Hint D.1 Introduction to Projectile Motion Part D How to approach the problem Hint not displayed Express your answer in seconds to two significant figures. ANSWER: = 1.0 Correct This situation, which you have dealt with before (motion under the constant acceleration of gravity), is actually a special case of projectile motion. Think of this as projectile motion where the horizontal component of the initial velocity is zero. Part E Imagine the ball on the left is given a nonzero initial speed in the horizontal direction, while the ball on the right continues to fall with zero initial velocity. What horizontal speed must the ball on the left start with so that it hits the ground at the same position as the ball on the right? Hint E.1 How to approach the problem Hint not displayed Express your answer in meters per second to two significant figures. ANSWER: = 3.0 Correct You can adjust the horizontal speeds in this applet. Notice that regardless of what horizontal speeds you give to the balls, they continue to move vertically in the same way (i.e., they are at the same y coordinate at the same time). Projectile Motion Tutorial Learning Goal: Understand how to apply the equations for 1-dimensional motion to the y and x directions separately in order to derive standard formulae for the range and height of a projectile. A projectile is fired from ground level at time initial speed Part A Find the time Hint A.1 , at an angle with respect to the horizontal. It has an . In this problem we are assuming that the ground is level. it takes the projectile to reach its maximum height. A basic property of projectile motion Hint not displayed Hint A.2 What condition applies at the top? Hint not displayed Hint A.3 Vertical velocity as a function of time Hint not displayed Hint A.4 Putting it all together Hint not displayed Hint A.5 A list of possible answers Hint not displayed Projectile Motion Tutorial Part A Express in terms of , , and (the magnitude of the acceleration due to gravity). ANSWER: = Correct Part B Find Hint B.1 Two possible approaches Hint not displayed Hint B.2 Some needed kinematics Hint not displayed Hint B.3 Solving for , the time at which the projectile hits the ground. Hint not displayed Express the time in terms of , , and . ANSWER: = Correct Part C Find , the maximum height attained by the projectile. Projectile Motion Tutorial Part C Hint C.1 Equation of motion Hint not displayed Hint C.2 When is the projectile at the top of its trajectory? Hint not displayed Hint C.3 Finding Hint not displayed Express the maximum height in terms of , , and . ANSWER: = Correct Part D Find the total distance (often called the range) traveled in the x direction; in other words, find where the projectile lands. Hint D.1 When does the projectile hit the ground? Hint not displayed Hint D.2 Where is the projectile as a function of time? Hint not displayed Hint D.3 Finding the range Hint not displayed Hint D.4 A list of possible answers Hint not displayed Projectile Motion Tutorial Part C Express the range in terms of , , and . ANSWER: = Correct The actual formula for is less important than how it is obtained: 1. Consider the x and y motion separately. 2. Find the time of flight from the y-motion 3. Find the x-position at the end of the flight - this is the range. If you remember these steps, you can deal with many variants of the basic problem, such as: a cannon on a hill that fires horizontally (i.e. the second half of the trajectory), a projectile that lands on a hill, or a projectile that must hit a moving target. A Wild Ride A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallel to the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the car along a certain section of the track is given by , where is a positive dimensionless constant. Part A At is the roller coaster car ascending or descending? Hint A.1 How to approach the problem Hint not displayed Hint A.2 Part A Find the vertical component of the velocity of the car Hint not displayed ANSWER: ascending descending Correct Part B Derive a general expression for the speed of the car. Hint B.1 How to approach the problem Hint not displayed Hint B.2 Magnitude of a vector Hint not displayed Hint B.3 Find the components of the velocity of the car Hint not displayed Express your answer in meters per second in terms of and . ANSWER: = Correct Part C The roller coaster is designed according to safety regulations that prohibit the speed of the from car exceeding . Find the maximum value of allowed by these regulations. Hint C.1 How to approach the problem Part C Hint not displayed Hint C.2 Find the maximum value of the speed Hint not displayed Express your answer using two significant figures. ANSWER: = 1.7 Correct Arrow Hits Apple An arrow is shot at an angle of above the horizontal. The arrow hits a tree a horizontal distance for the away, at the same height above the ground as it was shot. Use magnitude of the acceleration due to gravity. Part A Find Hint A.1 Find the initial upward component of velocity in terms of D. Hint not displayed Hint A.2 Find the time of flight in terms of the initial vertical component of velocity. Hint not displayed Hint A.3 Put the algebra together to find symbolically. , the time that the arrow spends in the air. Hint not displayed Answer numerically in seconds, to two significant figures. ANSWER: = 6.7 Correct Arrow Hits Apple Part A Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree. Part B How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as the arrow hits the tree? Hint B.1 When should the apple be dropped Hint not displayed Hint B.2 Find the time it takes for the apple to fall 6.0 meters Hint not displayed Express your answer numerically in seconds, to two significant figures. ANSWER: = 5.6 Correct Graphing Projectile Motion For the motion diagram given , sketch the shape of the corresponding motion graphs in Parts A to D. Use the indicated coordinate system. One unit of time elapses between consecutive dots in the motion diagram. Part A Construct a possible graph for x position versus time, . Hint A.1 Determine the initial value of Is the initial value of the x position positive, negative, or zero? ANSWER: positive negative zero Correct Hint A.2 Specify the shape of the graph Does the x position change at a constant rate or a changing rate? You can determine this by looking at the change in x coordinate from one dot to the next. ANSWER: Part A Hint A.1 Determine the initial value of Correct Since the x position changes at a constant rate (implying a constant x velocity), it must be represented by a graph with a constant slope. ANSWER: Part B Construct a possible graph for the y position versus time, . Hint B.1 Determine the initial value of Hint not displayed Hint B.2 Specify the shape of the graph Hint not displayed ANSWER: Part C Part B Construct a possible graph for the x velocity versus time, . Hint C.1 Determine the initial value of Is the initial value of the x velocity positive, negative, or zero? Look at the x component of the first arrow. ANSWER: Correct Hint C.2 Specify the shape of the graph Does the x velocity remain constant or does it change? You can determine this by comparing the x components of the arrows. ANSWER: Correct ANSWER: Part B Part D Construct a possible graph for the y velocity versus time, . Hint D.1 Determine the initial value of Hint not displayed Hint D.2 Specify the shape of the graph Hint not displayed Hint D.3 Specify the rate of change of Hint not displayed Part B ANSWER: Speed of a Softball A softball is hit over a third baseman's head with Part B speed and at an angle from the horizontal. Immediately after the ball is hit, the third baseman turns , for a time . He then around and runs straight back at a constant velocity catches the ball at the same height at which it left the bat. The third baseman was initially from the location where the ball was hit at home plate. Part A Find . Use for the magnitude of the acceleration due to gravity. Hint A.1 Find the initial velocity in the x direction Hint not displayed Hint A.2 Find the initial velocity in the y direction Hint not displayed Hint A.3 Find the total initial velocity Hint not displayed Express the initial speed in units of meters per second to four significant figures. ANSWER: = 18.77 Correct Part B Find the angle in degrees. Express your answer in degrees to four significant figures. ANSWER: 31.51 Answer Requested Part B Part C Find a vector expression for the velocity of the softball 0.1 s before the ball is caught. Hint C.1 vs is constant during the softball's motion, but Hint C.2 What is the equation for is a function of time. as a function of time Find ? Give your answer in terms of , and time . ANSWER: = Correct Hint C.3 Remember that Unit vectors is a projection of the velocty onto the vector, and velocty onto the vector. is a projection of the Use the notation , , an ordered pair of values separated by commas. Express your answer in units of meters per second to three significant figures. ANSWER: = 16.0,-8.82 Correct Part D Part B Part C Find a vector expression for the position of the softball 0.1 s before the ball is caught. Hint D.1 Equations of motion Hint not displayed Use the notation , , an ordered pair of values separated by commas, where and are expressed in meters, as measured from the point where the softball initially left the bat. Express your answer to three significant figures. ANSWER: = 30.4,0.932 Correct A Canoe on a River A canoe has a velocity of 0.510 southeast relative to the earth. The canoe is on a river that is flowing at 0.520 east relative to the earth. A Canoe on a River Part A Find the magnitude of the velocity of the canoe relative to the river. Hint A.1 How to approach the problem Hint not displayed Hint A.2 Let Find the relative velocity vector be the velocity of the canoe relative to the earth and the velocity of the water in the river relative to the earth. What is the velocity of the canoe relative to the river? Hint A.2.1 Relative velocity Hint not displayed ANSWER: Correct Hint A.3 Find the components of the velocity of the canoe relative to the river Let the x axis point from west to east and the y axis from south to north. Find and , the x and the y components of the velocity of the canoe relative to the river. Hint A.3.1 How to approach the problem Hint not displayed Hint A.3.2 Components of a vector A Canoe on a River Part A Hint A.3 Find the components of the velocity of the canoe relative to the river Hint not displayed Express the two velocity components, separated by a comma, in meters per second. ANSWER: , = -0.159,-0.361 All attempts used; correct answer displayed Now simply calculate the magnitude of , which is given by the square root of the sum of the squares of its components. Express your answer in meters per second. ANSWER: = 0.394 All attempts used; correct answer displayed Part B Find the direction of the velocity of the canoe relative to the river. Hint B.1 How to approach the problem The direction of a vector can be determined through simple trigonometric relations. You can use either the relation between the magnitude of the vector and one of its components or the relation between the two components of the vector. In both cases, use the information found in Part A. Note that the problem asks for the direction of as an angle measured south of west; your answer should be a positive angle between and . Hint B.2 Consider a vector of magnitude Find the direction of a vector given its components whose x component is and y component is . What is the angle this vector makes with the x axis? Hint B.2.1 The direction of a vector A Canoe on a River Part A Hint B.1 How to approach the problem Hint not displayed ANSWER: Correct Express your answer as an angle measured south of west. ANSWER: 66.2 Correct degrees south of west Score Summary: Your score on this assignment is 84.1%. You received 5.89 out of a possible total of 7 points. ... View Full Document