Explosions are like sticking collisions in reverse

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Explosions are like sticking collisions in reverse. Start with one body (often at rest), and it breaks into several moving pieces. Assuming F net =0 on the system, p 1 ,f + p 2 ,f + · · · = p i Common case: object at rest breaks into two moving objects: p 1 ,f + p 2 ,f = 0
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Phys 2A - Mechanics Example: An astronaut is “spacewalking” when he finds herself a few meters form the spaceship and decides it’s time to get back. She is not moving with respect to the ship and cannot walk. Her tether has broken so she cannot pull herself in. She decides to throw something away from the ship. But she knows she should not go too fast towards the ship. She can throw away a 2.0 kg- wrench from her toolbox, and she weighs 52.0 kg including suit and all tools. At what speed should she throw it if she wants to approach the ship at 1.0 m/s? ANS: Clearly F net =0 (no contact forces, no gravitational force). So she can use conservation of momentum. In the ship’s frame of reference (same as her’s, initially) her initial velocity is zero. So p 1 ,f + p 2 ,f = 0 where “1” refers to her and “2” to the tool. Now p 2 ,f = - p 1 ,f m tool v tool = - m she v she ,f (good luck!) | v tool | = m she m tool | v she ,f | = 52 - 2 2 . 0 (1 . 0) m/s = 25 m/s
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Phys 2A - Mechanics Example: Rocket motion. Rocket/jet propulsion works by throwing stuff like our astronaut of the previous example. But the “throwing” is continuous and at a fixed speed with respect to the rocket. There is an additional complication: as stuff is thrown the mass of the rocket decreases.
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Phys 2A - Mechanics Example: Rocket motion (continued). x y initial (time t) final (time t+ Δ t) m - Δ m Δ m m v v + Δ v v - u Here u = velocity of ejected stuff with respect to rocket (relative motion!) Conservation of momentum: P f = P i ( m - Δ m )( v + Δ v ) + Δ m ( v - u ) = mv or Neglecting Δ m Δ v (second order in small quantities) and simplifying m Δ v + u Δ m = 0 Divide by time interval, simplify and take limit Δ t ! 0 m Δ v Δ t = - u Δ m Δ t dv dt = - u 1 m dm dt rocket propulsion equation
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Phys 2A - Mechanics dv dt = - u 1 m dm dt The equation gives the acceleration of the rocket. It is positive (since the mass is decreasing in time).
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