A) Yes B) No C) Impossible to tell from the information given.

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1 Does escape speed depend on launch angle? That is, if a projectile is given an initial speed v o, is it more likely to escape an airless, non-rotating planet, if fired straight up than if fired at an angle? A) Yes B) No C) Impossible to tell from the information given. The speed of escape depends only on the relationship between the object s KE and PE at launch, which are independent of angle. L31 W 11/5/14 1

2 When thrust is turned off: E tot = KE + PE = 1 2 mv 2 0 G mm As long as the thrust is off, the rocket will have the same E tot. Its distance R will change, but then so will its speed v. R 0 (1) As R é, v ê. v 0 R 0 R E R E L31 W 11/5/14 2

3 When thrust is turned off: E tot = KE + PE = 1 2 mv 2 0 G mm As long as the thrust is off, the rocket will have the same E tot. Its distance R will change, but then so will its speed v. (1) As R é, v ê. (2) There is nothing in the conservation of energy about the angle at (3) which the rocket is fired. So, the angle doesn t matter. E tot = 0 Unbound Orbit Parabola v 0 = v escape KE E tot when < 0 thrust Bound is turned Orbit off = Circle, PE when Ellipse thrust v 0 is < turned v escape off R 0 = 2GM R 0 E tot > 0 Unbound Orbit Hyperbola v 0 > v escape L31 W 11/5/14 3

4 When thrust is turned off: E tot = KE + PE = 1 2 mv 2 0 G mm As long as the thrust is off, the rocket will have the same E tot. Its distance R will change, but then so will its speed v. (1) As R é, v ê. (2) There is nothing in the conservation of energy about the angle at (3) which the rocket is fired. So, the angle doesn t matter. E tot = 0 Unbound Orbit Parabola v 0 = v escape E tot < 0 Bound Orbit Circle, Ellipse v 0 < v escape KE E when tot > 0 thrust Unbound is turned Orbit off < Hyperbola PE when thrust v is 0 > turned v off escape R 0 = 2GM R 0 L31 W 11/5/14 4

5 When thrust is turned off: E tot = KE + PE = 1 2 mv 2 0 G mm As long as the thrust is off, the rocket will have the same E tot. Its distance R will change, but then so will its speed v. (1) As R é, v ê. (2) There is nothing in the conservation of energy about the angle at (3) which the rocket is fired. So, the angle doesn t matter. E tot = 0 Unbound Orbit Parabola v 0 = v escape E tot < 0 Bound Orbit Circle, Ellipse v 0 < v escape R 0 = 2GM R 0 E tot > 0 Unbound Orbit Hyperbola v 0 > v escape KE when thrust is turned off > PE when thrust is turned off L31 W 11/5/14 5

6 How about this: if a projectile is given an initial speed v o, is it more likely to escape an airless but rotating planet, if fired from the equator northward, northeastward, or eastward? It is most efficient to fire the projectile toward the east. Surface of the earth has an eastward rotational speed, v rot. v launch N v tot N NE E E v rot L31 W 11/5/14 6

7 Suppose a projectile is fired straight upward from the surface of an airless, nonrotating planet of radius R with the escape velocity v esc. What is the projectile's speed when it is a distance 4R from the planet's center (3R from the surface)? Hint: remember for the escape velocity E tot = 0 A) v esc B) v esc /2 C) v esc /3 D) v esc /4 E) v esc /9 0 = E tot = 1 2 mv2 G mm R v = 2GM R Therefore if R increases by a factor of 4, v reduces by a factor of 2. L31 W 11/5/14 7

8 Assignments Announcements: HW 10 is due this week. CAPA 11 is now live. You should read Ch. 10 now. Today: Finish up our discussion of orbits and gravity today. Will move on to rotational motion, found in Ch. 10. Also we ll discuss center-of-mass on Friday, found in Ch. 9. L31 W 11/5/14 8

9 To Create Artificial Gravity in Space centrifuge r x N=mg period τ Simulate Earth: N = mg What is the correct dynamical relationship between the centripetal force, N, and mg? A) mv2 r B) mv2 r C) mv2 r D) mv2 r = N = N mg = N + mg = 0 L31 W 11/5/14 9

10 To Create Artificial Gravity in Space Simulate Earth: N = mg centrifuge r x period τ mv 2 r v 2 = = N = mg circumference period 2 = 4π 2 r 2 τ 2 = gr N=mg L31 W 11/5/14 10

11 To Create Artificial Gravity in Space centrifuge r x N=mg period τ Simulate Earth: N = mg 4π 2 r 2 L31 W 11/5/14 11 τ 2 = gr To simulate earth s gravity, at what period must a wheel of radius r rotate in space? A) r / g B) g / r C) 2π r / g D) 2π g / r

12 To Create Artificial Gravity in Space centrifuge τ = 2π r / g r = 10 m x N=mg period τ Let r = 10 m, then τ 2π sec 6.3sec v = 2πr 2π (10m) τ 2π sec 10m / s L31 W 11/5/14 12

13 To Create Artificial Gravity in Space The astronaut drops his pen and observes it to fall to the floor. Which statement below is most accurate? A) After he releases the pen, the net force on the pen is zero. F net = 0 B) The pen falls because the centrifugal force pulls it toward the floor. C) The pen falls because the artificial gravity pulls it toward the floor. L31 W 11/5/14 13

14 To Create Artificial Gravity in Space Example: Spaceship Discovery 1 in the movie: L31 W 11/5/14 14

15 Rotational Motion -- rotation about a fixed axis. As distinct from translational motion which we have studied so far. What you ll see: We ll introduce variables to describe rotational motion (usually using Greek letters) and their relationships to one another will be exactly the same as for translational motion. The only change will be the symbols used and the meaning, of course. Translational Motion Rotational Motion position x (m) θ (rad) velocity v (m/s) = dx/dt ω (rad/s) = dθ/dt acceleration a (m/s 2 ) = dv/dt α (rad/s 2 ) = dω/dt L31 W 11/5/14 15

16 Rotational Motion -- rotation about a fixed axis. As distinct from translational motion which we have studied so far. What you ll see: We ll introduce variables to describe rotational motion (usually using Greek letters) and their relationships to one another will be exactly the same as for translational motion. The only change will be the symbols used and the meaning, of course. Constant acceleration a: Constant angular acceleration α: v = v 0 + at ω = ω 0 + αt x = x 0 + v o t at 2 v 2 = v aΔx θ = θ 0 + ω o t αt 2 ω 2 = ω αΔθ L31 W 11/5/14 16

17 Rotational Motion -- rotation about a fixed axis. As distinct from translational motion which we have studied so far. What you ll see: We ll introduce variables to describe rotational motion (usually using Greek letters) and their relationships to one another will be exactly the same as for translational motion. The only change will be the symbols used and the meaning, of course. Translational Motion Rotational Motion And, as we ll see, there are similar expressions for the rotational version of mass, force, Newton s laws, KE, conservation of Energy, momentum and its conservation, and so force. L31 W 11/5/14 17

18 Rotational Kinematics Describing rotational motion irrespective of rotational forces CCW: + Remember: θ s r (θ: rads) CW: - θ 0 = 0 or s = rθ Circumference = 2πr 360 = 2π rad 180 = π rad (r,θ) polar coordinates For circular motion, θ completely describes the motion. It gives the rotational position relative to a reference θ 0 = 0. L31 W 11/5/14 18

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