PH2201 Homework 5

Prof Marko B Popovic

Due in class on Tuesday January 31 2017

1. Student invented robotic sledge that can slide over ice with zero friction by throwing apples, i.e. 푚 one apple every second, ∆푡 = 1푠 , with speed 푣 = 10 relative to sledge reference frame. You may 푠 ignore mass of sledge and assume that mass of each apple is 푚푎 = 0.2 푘푔 . Initially there were 푁 = 500 apples on sledge. What is sledge’s distance travelled at the moment of last apple throw or alternatively after time 푡 = 푁∆푡. Assume that sledge started from rest. Do result depend on 푚푎? If instead of throwing one apple per second robotic sledge throws two apples every two seconds, is distance travelled different?

(Hint: Consider conservation of momentum for each apple throw. Mass of apples on sledge decreases after each apple throw but speed increases. Distance travelled here will be sum of sledge speeds, in reference frame of ice of course, for each time interval and then times ∆푡)

2. Consider now rocket which is not acted by any external force, i.e. an isolated body, in an abstract inertial frame, which utilizes similar principle as in previous problem. Instead of apples however rocket boosters generate exhaust with mass density 휌 that leave nozzle with speed 푣 relative to rocket. (For 푘푔 푚 example 휌 = 0.1 and 푣 = 300 .) 푚3 푠 (a) How quickly will rocket burn entire mass, 푀, of fuel during continuous steady operation if nozzle area is 퐴. What is distance travelled at the moment when last drop of fuel is utilized? (For example 푀 = 20000 푘푔 and 퐴 = 0.25 푚2.)

(b) Imagine now that rocket with full reservoir needs to overcome Earth gravity and be able to hover with zero speed relative to Earth just off the Earth’s surface. What is the minimal area 퐴푚푖푛of the nozzle? (Hint: Due to gravity the object, for example rocket, dropped from rest will increase speed by 푔∆푡 during first period of time ∆푡 and hence change in momentum will be 푀푔∆푡. How much exhaust moving with speed v in direction of gravity would one need to counteract that effect? Consider how much momentum that exhaust should carry on its own to “explain” effect of gravity.)

(c) What is the distance travelled in part (a) if minimal nozzle area (one that could resist Earth’s gravity, that you obtained in part (b)) is utilized (for rocket thought of as an isolated body)

3. Let’s derive Tsiolkovsky rocket equation for one dimensional case without gravity and solve it.

(a) Imagine moment in time 푡 when mass of the rocket is 푀(푡). If both rate of change of rocket mass 푑푀 < 0 and exhaust speed relative to rocket 푢 are constant and known quantities what is the speed of 푑푡 rocket at time 푡 , i.e. 푣(푡), if initial speed of rocket was zero and initial mass was 푀0? Express your answer 푑푀 in terms of 푡, 푀 , 푢, and 훽 = − . 0 푑푡 Start by writing equation of motion, 2nd Newton law for rocket. Label the force that exhaust pushes the 푑푣 rocket with 퐹(푡). You may express rocket acceleration as 푎 = 푑푡 Write the equation of motion , 2nd Newton law for exhaust. What is the force that rocket pushes exhaust?

Combine these two equations using the 3rd Newton law. Make sure you got all the signs right.

Now, you probably obtained differential equation with 푑푀 and 푑푣 terms. Group 푀 terms on same side with 푑푀 and 푣 terms (if any) on same side with 푑푣.

Multiply both sides with 푑푡 (you don’t need it) and integrate between initial and final state.

푑푙푛(푥) 1 푑푥 푑[푙푛(푥)+푐표푛푠푡] 1 It may help to know that = such that ∫ = 푙푛(푥). However be aware that = too, 푑푥 푥 푥 푑푥 푥 so you probably want to incorporate that const in your result and figure it out based on initial condition.

You should have now obtained Tsiolkovsky rocket equation that could help you figure out 푣(푡).

푑푀 Express now your answer in terms of 푡, 푀 , 푢, and 훽 = − . 0 푑푡 (b) What is 퐹(푡)? Is force changing throughout rocket motion or it is constant?

(c) What is travelled distance 푑(푡)?

푎−푏푥 푎 푎−푏푥 It may help to know that ∫ 푙푛 푑푥 = (푥 − ) [푙푛 ( ) − 1] + 푐표푛푠푡. 푎 푏 푎

푀 푀 (d) If you consider time 푡 = 0 would result for distance travelled 푑 (푡 = 0) be consistent with result in 훽 훽 problem 1?

4. Consider solar sail with mass 푀 which is essentially a large reflective mirror. Every time photon (quant of electromagnetic radiation, i.e. light) hits this mirror it gets reflected and transfer momentum to mirror. Assume that this momentum transfer equals twice the pre-collision photon momentum in mirror’s reference frame. If mirror is moving away from source of photons the light gets redshifted and appears in mirror’s reference frame as having less momentum then in original source reference frame. This can be expressed as

푐 − 푣푚푖푟푟표푟 푝푚푖푟푟표푟 = √ 푝푠표푢푟푐푒 푐 + 푣푚푖푟푟표푟

With c being speed of light and 푣푚푖푟푟표푟 speed of mirror in respect to source. Now imagine that source of photons is star that radiates net flux of photon momentum per unit perpendicular surface area at fixed distance of 1 AU (astronomical unit is average distance between Earth and Sun) every second, i.e.

푑2푃 푄(푟 = 1퐴푈) = 푑퐴푑푡 푘푔 You may assume that you know 푄(푟 = 1퐴푈) in units of . Because light spreads across the entire 푚푠2 surface area of sphere and because surface area is proportional to 푟2 you may assume that

푟2 푄(푟′) = 푄(푟 = 1퐴푈) 푟′2 (a) Write down the equation of motion for solar sail moving away from star in radial direction. Solar sail initial position was 푟 = 1퐴푈 and it started from rest.

(b) Solve this differential equation, more specifically express solar sail speed as function of radial distance from star. If solar sail goes very far from star, what will be its “terminal” speed?

(c) If someone ask you “How large should be the solar sail area 퐴 such that it can reach 푟 = 5퐴푈 in one (Earth’s) year?” what would be your response? You don’t need to actually solve this. Just describe how you would try to solve it in words. This should be clearly based on part (b).

푐−푣 HINTS: If you think it could simplify this problem you may Taylor expand √ 푚푖푟푟표푟 to first order in speed. 푐+푣푚푖푟푟표푟

푑푣 푑푟 푑푣 푑푣 You may find quite handy the following equality = = 푣 . Try to group terms with 푟 and 푣 on 푑푡 푑푡 푑푟 푑푟 opposite sides of equation before doing integration. You will then need the following integrals:

푓푖푛푎푙 ∫ 푑푓 = 푓푓푖푛푎푙 − 푓푖푛푖푡푖푎푙 푖푛푖푡푖푎푙

푓푖푛푎푙 푑푓 푓푓푖푛푎푙 ∫ = 푙푛 푖푛푖푡푖푎푙 푓 푓푖푛푖푡푖푎푙 푓푖푛푎푙 푑푓 1 1 ∫ 2 = − ( − ) 푖푛푖푡푖푎푙 푓 푓푓푖푛푎푙 푓푖푛푖푡푖푎푙 Keep in mind that you could change integration variable such that 푑푓 = 푑(푓 + 푐표푛푠푡). It could help too.

푐−푣 Finally, you may also find useful to Taylor expand 푙푛 in terms of 푣 ≪ 푐; however please don’t do it just 푐 to first order (as you will get silly result) but rather consider second order term too. Good luck!