230_Exam_2_Practice_Problems__Solutions - 230 Exam 2...

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230 Exam 2 Practice Problems: SolutionsJoseph BreenParticle Motion1.A particle moves along a path given by the curve:r(t) =2t2+ 1,2et+ 2, t3What are the normal and tangential components of acceleration when the particle is moving in thedirection of the y-axis?
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Projectile Motion1.One night, Joe sneaks onto the roof of Swift Hall — which is10meters tall — and throws his shoe atan angle of30degrees to the horizontal with an initial speed of2m/s. How fast is his shoe going whenit hits the ground? Take gravity to be10m/s2.Solution.Because this situation is implicitly two dimensional, and because there is no wind or anythingthat would give my shoe any extra acceleration, we can use the standard equations for projectile motion:r(t) =hx(t), y(t)i=x0+v0cos(θ)t, y0+v0sin(θ)t-12gt2In the context of this problem,g= 10,v0= 2, andθ=π6. Since I’m at the top of Swift Hall, we cantake my initial position to be(x0, y0) = (0,10). Plugging these values into our position equation givesus:r(t) =D3t,10 +t-5t2EIn order to calculate how fast the shoe is travelling when it hits the ground, we need to knowwhenthis happens. Well, the shoe hits the ground when theycomponent of our position vector is0, so:10 +t-5t2= 0Solving this fortgives ust1.5. To find the speed of the ball at this time, we first need to find thevelocity vector. This is just the derivative of the position:v(t) =hx0(t), y0(t)i=D3,1-10tESo the velocity vector at timet= 1.5is:v(1.5) =D3,-14EThus, the speed of the shoe when it hits the ground is:kv(1.5)k=q(3)2+ (-14)214.12.Joe is having a little too much fun on Dillo Day, makes a poor judgement call, and shoots a fireworkoff in front of Lunt Hall.The firework is launched in the western direction (in the direction of thenegativexaxis) at an angle of60degrees, with an initial speed of20m/s. The wind gives the fireworka northerly acceleration of3m/s2.Take gravity to be10m/s2.How far away from Lunt does the

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