(The spin of a baseball) When a ball is thrown with spin abo
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(The spin of a baseball) When a ball is thrown with spin about an axis that is not parallel to its velocity, it experien
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(The spin of a baseball) When a ball is thrown with spin about an axis that is not parallel to its velocity, it experiences a lateral acceleration due to differences in friction along its sides. This spin acceleration is given by as D kS v, where v is the velocity of the ball, S is the angular velocity of its spin, and k is a positive constant depending on the surface of the ball. Suppose that a ball for which k D 0:001 is thrown horizontally along the x-axis with an initial speed of 70 ft/s and a spin of 1,000 radians/s about a vertical axis. Its velocity v must satisfy 8 < : dv dt D .0:001/.1;000k/ v ? 32k D k v ? 32k v.0/ D 70i; since the acceleration of gravity is 32 ft/s2. (a) Show that the components of v D v1i C v2j C v3k satisfy 8 < : dv1 dt D ?v2 v1.0/ D 70 8 < : dv2 dt D v1 v2.0/ D 0 8 < : dv3 dt D ?32 v3.0/ D 0. (b) Solve these equations, and find the position of the ball t s after it is thrown. Assume that it is thrown from the origin at time t D 0. (c) At t D 1=5 s, how far, and in what direction, has the ball deviated from the parabolic path it would have followed if it had been thrown without spin?
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