HW #3-2-solutions

# HW #3-2-solutions - pasha(sep635 HW#3-2 Antoniewicz(56445...

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1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 (part 1 of 7) 2.0 points AstaroFmass7 × 10 30 kg is located at ~ r S = h 6 × 10 12 , 2 × 10 12 , 0 i m . Ap lane to Fma s s3 × 10 24 kg is initially lo- cated at ~ r P i = h 2 × 10 12 , 4 × 10 12 , 0 i m and is moving with an initial velocity oF ~ v P i = h 2000 , 16000 , 0 i m / s . At a time 1 × 10 6 slater ,whatisthenew velocity oF the planet? Start by fnding the x component, v P f,x . Correct answer: 2020 . 88 m / s. Explanation: In order to use the Momentum Principle to fnd the new velocity, we have to frst fnd the net gravitational Force on the planet From the star. The Formula For the gravitational Force be- tween two objects is ~ F gr = - Gm 1 m 2 | ~ r | 2 ˆ r. We know the masses oF the objects, so we just need to fnd the magnitude oF the distance between them and the unit vector pointing From the star to the planet. To fnd the vector pointing From the star to the planet (let’s call it ~ r P - S ), we subtract the position oF the star From that oF the planet: ~ r P - S = ~ r P - ~ r S = h 2 × 10 12 , 4 × 10 12 , 0 i m -h 6 × 10 12 , 2 × 10 12 , 0 i m = h- 4 × 10 12 , 2 × 10 12 , 0 i m . We need the length oF this vector and the unit vector pointing in the same direction. The length (or magnitude) is given by ± ± ± ~ r P - S ± ± ± = q ( - 4 × 10 12 m) 2 +(2 × 10 12 m) 2 =4 . 47214 × 10 12 m . And we use the magnitude to help us fnd the unit vector: ˆ r P - S = ~ r P - S | ~ r P - S | = h- 4 × 10 12 , 2 × 10 12 , 0 i m 4 . 47214 × 10 12 m = h- 0 . 894427 , 0 . 447214 , 0 i . Now we have everything we need to calcu- late the components oF ~ F gr .L e t sg oa h e a d and fnd all three components using vector algebra. ~ F gr = - Gm S m P | ~ r P - S | ˆ r P - S = - ( G )(7 × 10 30 kg)(3 × 10 24 kg) (4 . 47214 × 10 12 m) 2 ×h- 0 . 894427 , 0 . 447214 , 0 i = h 6 . 26412 × 10 19 , - 3 . 13206 × 10 19 , 0 i N , where G =6 . 67 × 10 - 11 m 3 · kg - 1 · s - 2 . Now that we know the net gravitational Force on the planet due to the star, we can use the Momentum Principle, Δ ~ p = ~ F net Δ t = ~ F gr Δ t, to fnd the new velocity, using the velocity update Formula we can derive From the Mo- mentum Principle: ~ p P f = ~ p P i + ~ F gr Δ t m P ~ v P f = m P ~ v P i + ~ F gr Δ t ~ v P

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HW #3-2-solutions - pasha(sep635 HW#3-2 Antoniewicz(56445...

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