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Republic of the Philippines
Department of Education
Regional Office IX, Zamboanga Peninsula
Z est for P rogress Z eal of P artnership
Science Grade 9
Quarter 4 - Module 2
Motion in Two Dimensions
Name of Learner: Grade & Section: Name of School:
Module 2 Motion in Two Dimensions What I Need to Know Hello, science enthusiasts! Have you been well? This module is written with you in mind. It is here to assist you in investigating the relationship between the angle of release and the projectile's height and range (S9FE-IVa-35). The scope of this module allows it to be used in many alternative learning situations. The language uses and recognizes the various terminology level of students. The lessons are set to follow the standard sequence of the course After going through this module, you're expected to:
- Describe the path of an object in projectile motion;
- Differentiate the horizontal and vertical motions of a projectile; and
- Explain the relationship between the angle of release and the height and range of the projectile. This lesson will discuss a type of motion in two-dimensions using projectile motion as an example. It focuses on the concept that two-dimension movements will be described and predicted using kinematics and dynamics. It also illustrates that true projectiles follow a parabolic path that is due to the downward pull of gravity. The activities also exhibit that the uniform horizontal motion (non-accelerated) is independent of the non- uniform (uniformly accelerated) vertical motion. What's In What is your favorite sport? Is it basketball, volleyball, badminton, or perhaps, ping pong? Have you ever joined Intramurals? When you throw a ball, how far can it travel? Or better yet, how hard do you need to serve for the volleyball to reach the other side of the court? Whether you're an athlete or a member of the cheering squad, you might have observed that the ball seems to be "flying" when thrown mid-air and appears to follow a specific path. Not only that, but you may have noticed that in many sports and games, players come "flying" too. These situations happen in real life and not only apply to sports but can also be used to track the path of meteorites and rockets' trajectories. How cool is that? In this module, you will be introduced to the concepts of understanding motion in two- dimensions that will help you employ the physics of sports and improve game events experiences.
II.Linear motion down an incline Release a ball on an inclined board. Draw and label the velocity-time and the acceleration-time graphs on the axes below. III. Two-dimensional along an incline A. Tracing the trajectory
- Make a marble launcher by attaching the popsicle sticks to the retractable pen and will serve as the launching pad of the marble. See Figure 1.
- On the board, select and draw fixed origins at points A and B. The left and bottom ends of the board may serve as the y-axis and x-axis, respectively. To complete the setup, elevate one end of the board using books with an angle of inclination of about 40◦. Get another book to hold the inclined surface, as shown in Figure 3. 3. Push the top end of the improved retractable pen and firmly hold it horizontally at point A. Then carefully place the powder-coated marble on its launching pad. Push the clip of the improved retractable pen to launch the marble. 4. Trace the path ( trajectory ) of the marble using a pencil. Label this path as "horizontally launched" and set aside later for analysis. Graph 3. Time graph for objects rolling straight down an incline Graph 4. Acceleration--- time graph for objects rolling straight down an incline Complete the sentence by encircling the answer. A ball rolling straight down an incline has a velocity that is ( increasing, decreasing ) as the object moves ( upward, downward ), and an acceleration that is ( constant, changing ) and ( upward, downward ). 40º Figure 2 Set up for projectile motion on an inclined plane Source: Science---Grade 9 Learner’s Module Figure 1 Retractable pen attached with popsicle as launching pad Source: Science---Grade 9 Learner’s Module
- Set the powder-coated marble on the launch pad at point B. Position the launching pad at the origin. Carefully launch the marble at 15º using the retractable pen.
- Trace the path (trajectory) with a pencil. Label this path as "launched at 15º angle."
- Do steps 5 and 6 for the other selected angles (30º, 45 º, 60º, and 75º). GUIDE QUESTIONS : (Please attach the graphing paper you used in the activity) Q1. Describe the path (trajectory) for horizontally-fired projectiles along an incline. Draw the path (trajectory) of the marble. Q2. Describe the form of the trajectory for projectiles fired at angles along an incline. Draw the path (trajectory) of the marble. Q3. Compare the locations of the trajectory peaks in terms of maximum height reached. Q4. Compare the horizontal distances (range) reached when they return to the elevation from which they were launched. Q5. Look at the path or the trajectories of projectiles fired at angles for the same launching velocity, which covered the greatest range (horizontal distance in the x-axis)? Q6. Among the trajectories of projectiles launched at angles, for the same launching speed, which recorded the highest peak? Q7. Which pairs of trajectories have almost equal ranges? DAY TWO ACTIVITY B. Recording the Hang Time
- Launch or project the marble at different angles on the inclined board. Record the hang time (using a stopwatch) of the marble from the time it was released until it hits the floor. Supply the table below using the data that you got in this activity. Safety check
- Ensure that the trajectories are free from hindrances. Table 1. Hangtime of the marble projected at different angles. Figure 3 Inclined illustration board supported between books for the marble projectile Source: Science---Grade 9 Learner’s Module
Figure 6 Matching trajectory A to a half parabola Source: Science---Grade 9 Learner’s Module of a projectile – how high it will travel, how far it will go, and so on. A notable thing to note is that the same range is obtained from two different projection angles– complementary angles. A body thrown into the air at an angle of 75 º , for example, will have a similar range as if it were thrown at the same speed at an angle of 15 º. An object thrown at 60 o^ will have the same range as when the object is launched at 30 o. As you can see, when we get the sum of 75o^ angle and 15o^ angle, 60o^ angle, and 30o^ angle, in both sets, we would obtain a 90 o^ angle. This means that the 75 o^ angle and 15 o^ angle are called complementary angles. Similarly, 60o^ angle and 30o^ angle are also complementary angles. Thus, complementary angles ( angles whose sum is equal to 90o ) would result in an equivalent range. Similarly, you have also observed that the marble, when launched at different angles, also reached different distances at different times. That is, for smaller angles, the object remains in the air for a shorter time. A maximum range is attained when an object is launched 45 o^ from the horizontal. What ' s More Activity 2: Curve A Like Objective: The students will be able to set a ball in projectile motion to match pre-drawn parabolic trajectories. Materials: Chalk manila paper (2 whole sheets) Small ball or any round object that is safe to throw (e.g., tennis ball, sepak takraw, etc.) Procedure:
1. Match-a-curve. a. Draw a rough parabola by drawing vertical and horizontal lines on manila paper and throw the ball like in Figure 6. GUIDE QUESTIONS: Q1. In what direction or orientation did you throw the ball? Q2. How would you describe the ball's path and motion? Q3. How many tries did you make to match the curved paths? b. Draw a box at the bottom end of the parabola. Throw the ball again with the box as the target. Q4. How many tries did you make before you matched the curves this time?
Figure 7 Matching trajectory B to a half parabola Source: Science---Grade 9 Learner’s Module Q5. What does this tell you regarding visuals or imaginary targets in sports?
1. What a curvy-a-throw! a. On another manila paper, draw a complete parabola and throw the ball similar to Figure 7. Q7. How would you describe the ball's path and motion? Q8. Aside from doing more trials or "practices" for this parabola, where will you place the imaginary target to aim at for better matching results? Q9. Based on the activity, is it possible that the ball will end at a higher elevation than its starting level? Q10. What force got the ball projected? Q11. What force continued to act on the ball when in mid-air? 3. Of curves a. The drawn curved graphs on the paper are parabolic curves. Similarly, trajectories A and B are also parabolic curves. Q12. How will you compare or contrast the horizontal and vertical spacing? Q13. What does the spacing in the set of vertical lines indicate about the vertical displacement and vertical velocity of the projectile motion?
- and arrows. The displacement, d, and velocity, v, are vector quantities. Projectile motion can be understood by analyzing the horizontal and the vertical components of the displacement and velocity, which add as vectors. Figure 8 Sketch of the velocity-vector component Source: Science---Grade 9 Learner’s Module
Step 4 : Get the answers. The horizontal velocity of the ball is 14.85 m/s. B. Projectiles Launched at an Angle If a projectile is launched upward at an angle, its velocity has two components:
- A constant horizontal velocity that travels in the same way as the launch, the acceleration of which is zero; and
- A rising positive vertical velocity component that is decreasing in magnitude until it becomes zero at the top of the trajectory (therefore, it no longer goes up any further). But because gravity makes it accelerates downward at a rate of 9.8 m/s per second or 9.8 m/s 2 , (therefore it stays at rest only for an instant), it will start to descend with an increasing negative vertical velocity until it is stopped by something. So as the projectile moves forward horizontally with uniform velocity, its vertical velocity is also accelerated, creating a trajectory that is a parabola. For full projectiles, objects are released at a certain angle from the horizontal. In this case, we can use the following equations to describe the motion of an object moving in full projectile motion. Where: θ = launch angle of the projectile, v 0 = initial velocity, and g =acceleration due to gravity SAMPLE PROBLEM: An athlete kicks a ball with an initial velocity of 25 m/s at an angle of 30o^ with the horizontal. Calculate the ball's time of flight, horizontal distance, and maximum height. Step 1 : Find what is required in the problem. Calculate the ball's time of flight, horizontal distance, and maximum height. Step 2 : Identify the given in the problem v = 25 m/s θ = 30o Step 3 : Use the half projectile motion equations to solve for the unknowns. x =
x = (25 m/s)^2 sin (2(30)) (9.8 m/s^2 x= (25 m/s)^2 sin (60) (9.8 m/s^2 x= 55. 23 m
Step 4 : Get the answers. The time of flight is 2.55 s, the maximum height is 7.97 m, and the horizontal range is 55.23 m. What I Have Learned For numbers 1-4, complete the sentence by encircling the best answer.
- A ( projectile, parabola ) is an object upon which the only force is gravity.
- Projectiles travel with a ( parabolic, straight ) trajectory due to the influence of gravity.
- When the initial launching angle is greater, the (shorter, longer) the range will be.
- If a ball is thrown at a 15-degree angle, it will have a (shorter, longer) range and height than a ball thrown at a 45-degree angle.
- What do you think?Two balls are set to move off a table. One is released, while the other is given an initial horizontal velocity. Which ball will reach the ground first? Explain.
**_What I Can Do_**
Projectile motion is a beneficial and practical concept in Physics. For example, if there are floods and rescuers could not reach the place, a rescue plane is usually used to drop a package of emergency rations to the victim. Can you think of some other ways how the concept of projectile motion is helpful in real-life situations?
II. Solve the following problems For number 1-3: A football player kicks a football from the ground level with an initial velocity of 27 m/s, 30 degrees above the horizontal.
- What is the maximum height (h) the ball attained? a. 2.44 m b. 7.89 m c. 9.30 m d. 20.
- How many seconds did it take the football to return to the launching height? a. 76 s b. 1.76 s c. 2.76 s d. 3.76 s
- How far away did it land (X)? a. 64.42 m b. 75.0 m c. 100.0 m d. 42.44 m For numbers 4-5: A physics book slides off a horizontal tabletop with a speed of 1. m/s. It strikes the floor in 0.350 s.
- What is the height of the table above the floor? a. 0.25 m b. 0.44 m c. 0.50 m d. 0.60 m
- What is the horizontal distance from the edge of the table to the point where the book strikes the floor? a. 0.30 m b. 0.39 m c. 1.0 m d. 1.5 m Additional Activities A bowling ball unintentionally falls out of an airliner's cargo bay as it flies along in a horizontal direction. As detected by a person standing on the ground and viewing the plane as in the figure at right, which path would the bowling ball most closely follow after leaving the airplane. ___________
Answer Key- Gr9Q4W2 Science What’s New Activity 1: Curve Me on an Incline Constant, zero I. Increasing, II. downward; constant, downward Activity 1: Curve Me on an Incline Guide Questions:
- The trajectory is a half open Q1: down parabola. Other students or down curve answer may concave down. . All the trajectories are full Q In parabolas. down- open addition, some students may also state something about different maximum heights, etc. . The trajectory peaks for each Q projection angle do not have the are peaks The location. same axis origin for- closest to the y shortest range or greatest angle is peak Each projection. of before half the range reached just indicates This travelled. was frictional forces between marble inclined surface and projectile
- resulting to a not so perfect open down parabola. . The trajectories have different Q horizontal distances (range) reached, short, quite are ranges some but some extend beyond the board or cookie sheet. . The trajectory fired closest to or Q ge. at 450 covered the greatest ran Q6. The trajectory with the greatest launching angle recorded the highest peak. have 0 and 75 0. Trajectories at 15 Q almost similar ranges. Trajectories at also have almost similar 0 and 60 0 30 0 but longer ranges than those for 15 students may note. Some 0 and 75 close ranges for pairs of angles that complementary not if almost are angles. The average range is longest. Q for the highest drop at 2 m and shortest at a 0.5 m height of fall.
What’s More
Activity 2: Curve Me on an Incline thrown was llba The. Q horizontally from the top curved is path ball’s The. Q drawn the to similar downwards moved it start, the At graph. horizontally forward but as it moved forward, it also moved downward. thrower’s the on Depends (. Q .) skills thrower’s the on Depends (. Q predictably lesser tries skills, but than before because of the visual goal.)
. Aiming at visual goals makes Q practice easier and results in better approximations of flight. . The ball was thrown upward Q n angle from from the bottom left at a horizontal. . The ball moved up in a curved Q path until it reached a maximum height and then it moved downward still following the curved path. an have to best is It. Q imaginary target at the top of the anywhere else curve rather than along the parabola. . In both throws the balls Q lower a on up end always elevation. It is not possible that the ball will end at a higher elevation than its starting level. . The initial push from the Q throw. d. The force of gravity acte Q at all times on the ball. between spacing The. Q horizontal lines is equal unlike vertical between spacing the the by increases which lines square of a span/unit. distance increasing The. Q between vertical lines indicate is motion vertical the that accelerated due to gravity. What I Have Learned Projectile 1. Parabolic 2. Shorter 3. Shorter 4. Both balls will reach the 5. ground at the same time since both balls are acted by the same gravitational force What’s I Can Do Answers may vary. **Assessment II. I.
- C B 1.
- C B 2.
- A D 3.
- D C 4.
- B C 5. A 6. B 7. C 8. C 9. D 10.**
