Transcript Review
Chapter 10: Projectile and Satellite Motion Brandon Hartfiel Homework due 3/21 exercises 44 45 46 52 53 54 problems 4 5 6 10 Review Theory of Universal Gravitation Near the earth’s surface Gm1m2 f 2 d f mg Vertical motion due to gravity near the earth’s surface g=9.8 ~ 10 1 2 y at v0t y0 2 example fall off cliff Projectile Motion What happens if we throw an object sideways? We can calculate the vertical and horizontal components separately Vertical motion Horizontal motion 1 2 y a y t v0 y t y0 2 x v0 xt x0 There is no acceleration in the horizontal direction figure 10.4, run off cliff What happens if we turn gravity off? Gravity on y v0 yt y0 1 2 y a y t v0 y t y0 2 Gravity off y v0 yt y0 y v0 y t y 0 this is the only difference So we can just draw what would have happened without gravity, then subtract 5t2 from the height. see fig 10.8 Interesting things about projectiles launched from the ground • time up = time down • speeds are equal at points of equal height • shape is a parabola y=ax2+b • maximum horizontal distance is obtained at 45 degrees • complementary angles (a+b=90) give same distance figure 10.11 and now for something completely different … Planetary Motion • Ptolemy (90-168) – earth centered universe all motion circular epicycles used to explain retrograde motion retrograde motion • Copernicus (1473-1543) – Earth is in the center – no epicycles. • Galileo (1564-1642) – Used telescope to observe the phases of Venus. Phases of the inner planets Johannes Kepler (1571-1630) • Noticed circles didn’t work for Mars orbit. • After trying about 40 other shapes, he found that the orbit was an ellipse and that the sun was at one of the foci. (Kepler’s first law) 2 equation for an ellipse 2 x y 1 2 2 a b If a=b, you have a circle with radius a and both foci are in the middle. So perfectly circular orbits are possible too • He also found that the line between the sun and a planet sweeps out equal areas in equal intervals of time. Kepler's Laws So planets move faster when they are closer to the sun. • This is what we expect from conservation of energy (invented after Kepler) Close to the sun – high kinetic energy low gravitational potential energy Far from the sun – low kinetic energy high gravitational energy • By looking at the other planets, Kepler found that the period, P, of a planet’s orbit (length of a year) is related to its “average”* distance from the sun, r, in the following way P2=kr3 where k is some constant. Solar System Viewer * actually it’s the length of the ellipse’s semi major axis Isaac Newton (1643-1727) start with Kepler’s third law P kr 2 Period=circumference of orbit/velocity = P=2pd/v v2 from plug in the centripetal force equation F=mv2/r (pg. 144) 22 p 2 r 2 3 kr v2 22 p 2 r 2 m kr3 Fr 22 p 2 m k 2 Fr cancel extra factors of r rearrange 3 22 p 2 m F kr 2 Force is proportional to 1/r2 1/r2 Maybe gravity is responsible for the planets’ orbits and the motion of projectiles on earth • But why do planets move in ellipses and projectiles in parabolas? When we calculate a projectile’s path near the surface of the earth, we use g=9.8 , but because g is inversely proportional to the distance from the center of the earth squared, the real value is 2 (9.8)(6,400,000) 2 (height 6,400,000) over short distances, there isn’t much difference, but if we calculate the exact path of a projectile it is an ellipse too. draw ellipse and parabola on the board, draw 5m vs 8k, figure 10.33