. Differential and integral calculus, an introductory course for colleges and engineering schools. he particle are obtained by substituting v0 = 2 m sin \ 0, and tf> = - (*■ — 0) in equations (2) of Art. 153, z and are x = 2msin2^0-^, y = 2msin£0cos£0»£ — %gt2, which reduce to x = m [1 + cos (tt — 0)] t, y = m sin (tt — 0) • t — \ gt2. These last equations show that the motion may be conceived as com-posed of two motions: First, a translation in the positive direction of OX with a velocity m. Second, a projection from Pwith a velocity m in the directionof the tangent to the wheel at
. Differential and integral calculus, an introductory course for colleges and engineering schools. he particle are obtained by substituting v0 = 2 m sin \ 0, and tf> = - (*■ — 0) in equations (2) of Art. 153, z and are x = 2msin2^0-^, y = 2msin£0cos£0»£ — %gt2, which reduce to x = m [1 + cos (tt — 0)] t, y = m sin (tt — 0) • t — \ gt2. These last equations show that the motion may be conceived as com-posed of two motions: First, a translation in the positive direction of OX with a velocity m. Second, a projection from Pwith a velocity m in the directionof the tangent to the wheel at results can also be ar-rived at in the following way: If the wheel did not rotate butmerely slipped along the roadwith a speed m, the particle at Pwould have simply a horizontalO motion with a speed m. If, on the other hand, the wheel were to turn on its axle with a (tangential)velocity m, while the axle itself remained stationary, the particle~ at Pwould have a velocity m along the tangent to the wheel at P. And theactual motion is obviously compounded of these two §156 APPLICATIONS OF INTEGRATION IN KINEMATICS 223 156. Exercises. 1. If the wheels of a bicycle are 3 feet in diameter, and are without mudguards, and if the wheel is driven at a speed of 15 miles per hour, what isthe highest flight of mud from the tires, and at what angle must it bethrown off to reach this height? 2. A body has an initial velocity v0 and is subjected to a resistancethat at any instant is proportional to its velocity at that instant. Theequation of motion is then d2s dv ~, w> = Jt = -Kv- Determine the motion completely. This is approximately the law of re-sistance offered by air or water to a body whose velocity does not exceed10 or 12 feet per second. 3. To a falling body, such as a raindrop, the air offers a resistancethat at any instant is proportional to the square of the velocity at thatinstant. The equation of motion is therefore d2s dv „„ 2 Determine the
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