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# Math Project

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### Math Project

1. 1. Math Project!  <br />Rachel Carrillo<br />Marian Hsiung<br />Natalia Castro<br />Holly Qiu<br />Period A & 5<br />May, 2010<br />
2. 2. Problem<br />Pg. 162 #14<br />An astronaut standing on the moon throws a rock into the air, the height of the rock is<br /> s(t)= -(27/10)t2 + 27t + 6<br />Where s(t) is measured in feet and t is measured in seconds.<br />a.) Find the expression for the velocity and acceleration of the rock.<br />b.)Find the time when the rock is at its highest point by finding the time when the velocity is zero. What is the height of the rock at this time?<br />c.)How does the acceleration of the rock compare with the acceleration due to gravity of earth?<br />
3. 3. a.) Find the expression for the velocity and acceleration of the rock.<br />Original equation: s(t)= -(27/10)t2 + 27t + 6<br /> s’’(t) = v’(t) = a(t)<br />V(t)= s’(t)= -(27/10)(2)t + 27(1) = 6(0)<br />= -(27/5)t + 27 in ft/sec<br />A(t)= s’’(t)= -(27/5) ft/sec2 (constant acceleration)<br />
4. 4. b.)Find the time when the rock is at its highest point by finding the time when the velocity is zero. What is the height of the rock at this time?<br />-(27/5)t + 27 =0 t=5<br />S(5)=73.5 ft<br />
5. 5. c.)How does the acceleration of the rock compare with the acceleration due to gravity of earth?<br />The gravity of earth is -32.1740 ft/s2<br />The gravity of the moon is –(27/5) ft/s2<br />The acceleration of the moon is less than the Earth’s therefore, the gravity pull on earth is stronger.<br />
6. 6. What causes Gravity to differ?<br />Gravity is the attractions between objects. It is directly correlated with mass.<br />Mass:<br /> Jupiter > Earth > Moon Therefore<br />Gravity: <br /> Jupiter > Earth > Moon<br />
7. 7. Across the Universe<br />By finding the gravity/acceleration of each planet and integrating it, given an initial condition, we can predict the maximum height the rock would reach if thrown in every planet.<br />∫ A(t) dt= V(t)<br />∫ V(t) dt= S(t)<br />Given:<br />The rock is thrown with an initial velocity of 10 ft/sec.<br />The astronaut is 6 ft high.<br />
8. 8. Mercury<br />Gravity: 12.22 ft/s2<br />A(t)=12.22<br />∫ A(t) dt= 12.22t+C=V(t)<br />V(0)=10<br />V(t)= 12.22t+10<br />∫V(t) dt= 6.11t2 +10t+C=S(t)<br />S(0)= 6<br />S(t)= 6.11t2 +10t+6<br />
9. 9. Venus<br />Gravity: 28.2ft/s²<br />A(t)=28.2<br />∫ A(t) dt=28.2t+C=V(t)<br />V(0)=10<br />V(t)=28.2t+10<br />∫V(t) dt=14.1t²+10t+C=S(t)<br />S(0)=6<br />S(t)=14.1t²+10t+6<br />
10. 10. Mars<br />Gravity: 12.22 ft/s2<br />A(t)=12.22<br />∫ A(t) dt= 12.22t+C=V(t)<br />V(0)=10<br />V(t)= 12.22t+10<br />∫V(t) dt= 6.11t2 +10t+C=S(t)<br />S(0)= 6<br />S(t)= 6.11t2 +10t+6<br />
11. 11. Jupiter<br />Gravity: 81.75 ft/s²<br />A(t)=81.75<br />∫ A(t) dt= 81.75t+C=V(t)<br />V(0)=10<br />V(t)=81.75t+10<br />∫V(t) dt=40.875t²+10t+C=S(t)<br />S(0)=6<br />S(t)=40.875t²+10t+6<br />
12. 12. Saturn<br />Gravity: 10.584 ft/s²<br />A(t)=10.584<br />∫ A(t) dt=10.584t+C=V(t)<br />V(0)=10<br />V(t)=10.584t+10<br />∫V(t) dt=5.292t²+10t+C=S(t)<br />S(0)=6<br />S(t)=5.292t²+10t+6<br />
13. 13. Uranus<br />Gravity: 28.2ft/s²<br />A(t)=28.2<br />∫ A(t) dt=28.2t+C=V(t)<br />V(0)=10<br />V(t)=28.2t+10<br />∫V(t) dt=14.1t²+10t+C=S(t)<br />S(0)=6<br />S(t)=14.1t²+10t+6<br />
14. 14. Pluto (YES, it IS a planet)<br />Gravity: 25.722 ft/s²<br />A(t)=25.722<br />∫ A(t) dt=25.722t+C=V(t)<br />V(0)=10<br />V(t)=25.722t+10<br />∫V(t) dt=12.861t²+10t+C=S(t)<br />S(0)=6<br />S(t)=12.861t²+10t+6<br />
15. 15. Another Example is…..<br />
16. 16. Problem with Weight and Height<br />Pg. 134, number 35<br />Problem: To estimate the height of a building, a weight is dropped from the top of the building into a pool at ground level. How high is the building if the splash is seen 9.2 seconds after the weight is dropped?<br />
17. 17. Answer<br />S(t)= -16t2 + s0<br />S(9.2)= -16(9.2)2 + s0<br /> s0 =1354.24 ft<br />