1. A rock is thrown at an angle q to the horizontal. Ignore air resistance. a) S
ID: 1705376 • Letter: 1
Question
1. A rock is thrown at an angle q to the horizontal. Ignore air resistance.a) Starting from Newton’s second law, find the location of the rock as a function of time in horizontal and vertical Cartesian coordinates.
b) If r(t) is the distance the rock is from its starting point, for what initial angles q will r(t) increase monotonically as the rock flies through the air? (Suggestion: To avoid long square roots, write out an expression for r^2, using the results of part a), and consider whether it has a maximum).
Explanation / Answer
a) we have v_ver=v0*sinp. v_hor=v0*cosp. so y=v0*sinp*t-gt^2/2 x=v0*cosp*t. b) r(t)=sqrt(y^2+x^2). r(t)=sqrt((v0*sinp*t-gt^2/2)^2+(v0*cosp*t)^2)=sqrt(v0^2*t^2+g^2*t^4/4-v0*sinp*g*t^3) at t_max=v0sinp*2/g --- in order to see if r(t) increased mono, so at t=t_max, r^2 has a maximum. take the derivative of r^2 to t. v0^2*2*t+g^2*t^3-3v0*sinp*g*t^2=0. plug 2v0*sinp/g=t_max in. 4*v0^3*sinp/g+8*v0^3*sinp^3/g-12*v0^3*sinp^2/g=0. simplify. 4+8*sinp^2-12*sinp=0. solve this. sinp=1 or 1/2. 1 is wrong. so sinp=1/2 so p=30.
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