Concept explainers
A space probe describes a circular orbit of radius nR with a velocity v0 about a planet of radius R and center O. Show that (a) in order for the probe to leave its orbit and hit the planet at an angle θ with the vertical, its velocity must be reduced to αv0, where
(b) the probe will not hit the planet if α is larger than
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Vector Mechanics for Engineers: Statics and Dynamics
- 2B Show that in a field of attractive forces F(r) a body of mass m can always perform circular motion of radius r0 with constant angular velocity w. Also show that the velocity u of the circular orbit is given by the relation u² == ToF (1) marrow_forwardChannel AB is fixed in space, and its centerline lies in the xy plane. The plane containing edges AC and AD of the channel is parallel to the xz plane. The surfaces of the channel are frictionless and the sphere E has 1.9 kg mass. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. N N E 30° x F 20° B ᎠᏓ C 30°/A 30° Determine the force supported by cord EF, and the reactions RC and RD between the sphere and sides C and D, respectively, of the channel. (Round the final answers to four decimal places.) The force supported by cord EF is The reactions RC and Rp between the sphere and sides Cand D. respectively, of the channel are as follows: RC= RD= z N. 4arrow_forwardA spacecraft approaching the planet Saturn reaches point A with a velocity vA of magnitude 68.8 × 103 ft/s. It is to be placed in an elliptic orbit about Saturn so that it will be able to periodically examine Tethys, one of Saturn’s moons. Tethys is in a circular orbit of radius 183 × 103 mi about the center of Saturn, traveling at a speed of 37.2 × 103 ft/s. Determine (a) the decrease in speed required by the spacecraft at A to achieve the desired orbit, (b) the speed of the spacecraft when it reaches the orbit of Tethys at B.arrow_forward
- The total momentum of a system of masses (moving bodies) in any one direction remains constant unless acted upon by an external force in that direction". This principle is applied to problems on collision of two bodies. Select one: O True O Falsearrow_forwardM XCM M A binary system is shown by the image above. It consists of two stars of equal mass. These stars revolve in a circular orbit about thgeir center of mass, which is midway between them. If the orbital speed of each star is 2,280 km/s and the orbital period of each is 11.7 days. Find the mass M of each star.arrow_forwardVA₂ PLANE OF CONTACT ÚB₂ X A B 0 m m velocity = 1.52 On an air hockey table, two pucks of identical mass collide in the middle of the rink. Puck A had an initial and puck B had an initial velocity B = 3.1 according to the x-y axis shown. The plane of contact can be thought of as a line angled 0 = 32° above the x-axis. If the coefficient of restitution is e = 0.41 between the pucks, what are the velocities of the pucks after the impact? s i+ i+ m S m Ⓒ 0 UBC Engineering Sarrow_forward
- A satellite is in a circular earth orbit of radius I'min = 1.67R, where R is the radius of the earth. What is the minimum velocity boost Av necessary to reach point B, which is a distance max = 2.68R from the center of the earth? At what point in the original circular orbit should the velocity increment be added? Answer: Av= i min max B m/sarrow_forwardA space probe is to be placed in a circular orbit of 5600-mi radius about the planet Venus in a specified plane. As the probe reaches A, the point of its original trajectory closest to Venus, it is inserted in a first elliptic transfer orbit by reducing its speed by ΔvA. This orbit brings it to point B with a much reduced velocity. There the probe is inserted in a second transfer orbit located in the specified plane by changing the direction of its velocity and further reducing its speed by ΔvB. Finally, as the probe reaches point C, it is inserted in the desired circular orbit by reducing its speed by ΔvC. Knowing that the mass of Venus is 0.82 times the mass of the earth, that rA = 9.3 × 103 mi and rB = 190 × 103 mi, and that the probe approaches A on a parabolic trajectory, determine by how much the velocity of the probe should be reduced (a) at A, (b) at B, (c) at C.arrow_forwardA spacecraft of mass m describes a circular orbit of radius ị around the earth. (a) Show that the additional energy AE that must be imparted to the spacecraft to transfer it to a circular orbit of larger radius r, is GMm(r2 – r¡) ΔΕ= where M is the mass of the earth. (b) Further show that if the transfer from one circular orbit to the other is executed by placing the space- craft on a transitional semielliptic path AB, the amounts of energy AE, and AEg which must be imparted at A and B are, respectively, proportional to r, and r¡: ΔΕΞ ΔΕΔΕ, ΔΕarrow_forward
- A small ball swings in a horizontal circle at the end of a cord of length l1 , which forms an angle 01 with the vertical. The cord is then slowly drawn through the support at O until the length of the free end is l2. (a) Derive a relation among l1, l2, 01, and 02. (b) If the ball is set in motion so that initially l1 = 0.8 m and 01 = 35°, determine the angle 02 when l2= 0.6 m.arrow_forwardA satellite is in a circular earth orbit of radius min = 1.66R, where R is the radius of the earth. What is the minimum velocity boost Av necessary to reach point B, which is a distance max = 3.94R from the center of the earth? At what point in the original circular orbit should the velocity increment be added? CO Answer: Av = i max m/sarrow_forwardGravitational Slingshot Often in designing orbits for satellites, people use what is termed a "gravitational slingshot effect." The idea is as follows: A satellite of mass m; and speed v,i circles around a planet of mass m, that is moving with speed v in the opposite direction. See the diagram below: Although the satellite never touches the planet, this interaction can still be treated as a collision because of the gravitational interaction between the planet and satellite during the slingshot. Since gravity is a conservative force, the collision is elastic. Use an x-axis with positive pointing to the right. Solve for the unknowns below algebraically first, then use the following values for the parameters. m, = 2.40E+24 kg m; = 880 kg Viz = 3.050E+3 m/s Vpiz = -6.10E+3 m/s Solve for the final velocity of the satellite after the collision.arrow_forward
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