Orbital Motion SE Answer Key

Student Exploration: Orbital Motion – Kepler’s Laws Vocabulary: astronomical unit, eccentricity, ellipse, force, gravity, Kepler’s first law, Kepler’s second law, Kepler’s third law, orbit, orbital radius, period, vector, velocity Prior Knowledge Questions (Do these BEFORE using the Gizmo.) 1. The orbit of Halley’s Comet, shown at right, has an oval shape. In which part of its orbit do you think Halley’s Comet travels fastest? Slowest? Mark these points on the diagram at right. 2. How might a collision between Neptune and Halley’s Comet affect Neptune’s orbit? It probably wouldn't have a massive effect on neptune orbit because neptune is bigger than the comet Gizmo Warm-up The path of each planet around the Sun is determined by two factors: its current velocity (speed and direction) and the force of gravity on the planet. You can manipulate both of these factors as you investigate planetary orbits in the Orbital Motion – Kepler’s Laws Gizmo. On the CONTROLS pane of the Gizmo, turn on Show trails and check that Show vectors is on. Click Play ( ). 1. What is the shape of the planet’s orbit? oval 2. Watch the orbit over time. Does the orbit ever change, or is it stable? it doesn't change 3. Click Reset ( ). Drag the tip of the purple arrow to shorten it and reduce the planet’s initial velocity. Click Play. How does this affect the shape of the orbit? It becomes smaller and more flat Get the Gizmo ready: Activity C: ● Click Reset. Radius and period ● Set the Sun mass to medium, close to the mass of our Sun. Introduction: Kepler’s last law relates the period (T) of a planet, or the time it takes to complete an orbit, to the planet’s orbital radius (a), or average distance from the Sun. Question: How does the period of a planet relate to its average distance from the Sun? 1. Predict: How do you expect the period of a planet to change as its average distance from the Sun increases? itll become longer 2. Measure: Place the planet at -4.00i AU and set the velocity to -15.00j km/s (does not have to be exact). Click Play, and then click Pause when the planet completes one orbit. A. What is the approximate period of the planet in days? I got about 2,035 days B. Select the TABLE tab and click Record data. What is the actual period (T) of the planet in Earth days and years? 15 days 5 years C. What is the orbital radius (a) of the planet? 2.49 D. Click Reset. Place the planet at -2.00i AU and set the velocity to -20.0j km/s. Click Play and Record data. What is the period now? 188 days 1 year E. What is the new orbital radius? 1.11 F. How does the distance between the planet and the Sun affect its period? As distance increases, the period decreases 3. Gather data: Experiment with the Gizmo to create a series of larger and larger orbits. Record each orbit’s radius (a) and period (T). Leave the last two columns blank for now. Orbit a (AU) T (years) a^3 t^2 1 .47 .453 .1038 .2059 2 1.11 1.638 1.367 2.683 3 1.76 3.273 5.451 10.712 4 2.49 5.511 15.438 30.371 5 3.31 8.460 36.264 71.516 (Activity C continued on next page) Activity C (continued from previous page) 4. Analyze: How does the period change as the orbital radius increases? The period increases 5. Interpret: Select the GRAPH tab, and check that the graph of T vs. a is selected. Use the zoom controls to adjust the graph so you can see all your data and it fills the graph. A. What does the graph indicate? period vs orbital radius B. Does the graph of T vs. a form a straight line? no 6. Investigate: Choose other options in the dropdown menus until you find a graph that makes a perfectly straight line. (If you cannot see all of your data, click the [–] controls to zoom out.) Which relationship makes a straight line? T^2 and A^3 When you have a line, click the camera ( ) icon to take a snapshot. Right-click the image, and click Copy Image. Paste it into a blank document you will turn in with this worksheet. 7. Calculate: On the table on the previous page, label the third column “a 3” and the fourth column “T 2.” Use a calculator to cube each value of a and square each value of T. What do you notice? As A gets higher, A^3 gets higher- same with T and T^2 Kepler’s third law states that the cube of a planet’s orbital radius is proportional to the square of a planet’s period: a3 = kT 2 for some constant k. If the radius is measured in astronomical units, the period is measured in Earth years, and the mass of the star is equal to the mass of our Sun, the value of k is equal to 1 AU 3/y2. 8. Challenge: The orbital radii of the planets are given in the table below. Calculate the period of each planet (in Earth years), and then check your values with your teacher. Planet a (AU) T (years) Planet a (AU) T (years) Mercury 0.39 .240 Jupiter 5.20 11.862 Venus 0.72 .615 Earth 1.00 1 Saturn 9.54 29.457 Uranus 19.18 Mars 1.52 1.881 Neptune 84.021 30.06 164.8