2. (From an old exam) A toy train travels counterclockwise around a circular track of radius R. Taking the center of the circle as the origin, the position vector of the train as a function of time is F(t) = R Cos (wt)i + Rsin (wt)ĵ where is a constant. a) Find an expression for the train's velocity vector as a function of time. b) Find an expression for the train's speed at any given time. Simplify it as much as you can. (Recall that cos² 0 + sin² 0 = 1, for any 8.)

2. (From an old exam) A toy train travels counterclockwise around a circular track of radius R. Taking the center of the circle as the origin, the position vector of the train as a function of time is F(t) = R Cos (wt)i + Rsin (wt)ĵ where is a constant. a) Find an expression for the train's velocity vector as a function of time. b) Find an expression for the train's speed at any given time. Simplify it as much as you can. (Recall that cos² 0 + sin² 0 = 1, for any 8.)

An Introduction to Physical Science

14th Edition

ISBN:9781305079137

Author:James Shipman, Jerry D. Wilson, Charles A. Higgins, Omar Torres

Publisher:James Shipman, Jerry D. Wilson, Charles A. Higgins, Omar Torres

Chapter2: Motion

Section: Chapter Questions

Problem IM

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Review. You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the x axis and at a fixed height of 7.60 103 m. At time t = 0, the airplane is directly above you so that the vector leading from you to it is P0=7.60103m. At t = 30.0 s, the position vector leading from you to the airplane is P30=(8.04103i+7.60103j)m as suggested in Figure P3.31. Determine the magnitude and orientation of the airplanes position vector at t = 45.0 s. Figure P3.31
Peter throws a baseball through a houses window. He stands 3.4 m from the window. The ball starts in his hand 1.1m above the ground and goes through the window at a point 2.4 m off the ground. a. Sketch the problem, including an xy coordinate system with the origin on the ground directly below the initial position of the ball, the y axis pointing upward, and the x axis pointing toward the house. b. What are the initial and final coordinates of the ball?
(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40.0° north of east (which is equivalent to subtracting B from A that is, to finding R=AB ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40.0° south of west and then 12.0 m in a direction 20.0° east of south (which is equivalent to subtracting A from B that is, to finding R=BA=R ). Show that this is the case.
The vector position of a particle varies in time according to the expression r=3.00i6.00t2j, where r is in meters and t is in seconds. (a) Find an expression for the velocity of the particle as a function of time. (b) Determine the acceleration of the particle as a function of time. (c) Calculate the particles position and velocity at t = 1.00 s.
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure P6.3. The length of the arc ABC is 235 m, and the car completes the turn in 36.0 s. (a) What is the acceleration when the car is at B located at an angle of 35.0? Express your answer in terms of the unit vectors i and j. Determine (b) the cars average speed and (c) its average acceleration during the 36.0-s interval. Figure P6.3
You can use any coordinate system you like to solve a projectile motion problem. To demonstrate the truth of this statement, consider a ball thrown off the top of a building with a velocity v at an angle with respect to the horizontal. Let the building be 50.0 m tall, the initial horizontal velocity be 9.00 m/s, and the initial vertical velocity be 12.0 m/s. Choose your coordinates such that the positive y-axis is upward, the x-axis is to the right, and the origin is at the point where the ball is released, (a) With these choices, find the balls maximum height above the ground and the time it takes to reach the maximum height. (b) Repeat your calculations choosing the origin at the base of the building.
The rectangle shown in Figure P3.56 has sides parallel to the x and y axes. The position vectors of two corners are = 10.0 m at 50.0 and = 12.0 m at 30.0. (a) Find the perimeter of the rectangle. (b) Find the magnitude and direction of the vector from the origin to the upper-right corner of the rectangle.
A map suggests that Atlanta is 730 miles in a direction 5.00c north of east from Dallas. The same map shows that Chicago is 560 miles in a direction 21.0 west of north from Atlanta. Figure P3.18 shows the location of these three cities. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago. Figure P3.18
You fly 32.0 km in a straight line in still air in the direction 35.0° south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction 45.0° south of west and then in a direction 45.0° west of north. These are the components of the displacement along a different set of axes—one rotated 450°.
Figure P3.60 illustrates the difference in proportions between the male (m) and female (f) anatomies. The displacements d1m. and d1f from the bottom of the feet to the navel have magnitudes of 104 cm and 84.0 cm, respectively. The displacements d2m and d2f have magnitudes of 50.0 cm and 43.0 cm, respectively. (a) Find the vector sum of the displacements dd1 and dd2 in each case. (b) The male figure is 180 cm tall, the female 168 cm. Normalize the displacements of each figure to a common height of 200 cm and re-form the vector sums as in part (a). Then find the vector difference between the two sums. Figure P3.60
Prove that the trajectory of a projectile is parabolic, having the form y=ax+bx2. To obtain this expression, solve the equation x=v0xt for t and substitute it into the expression for y=v0yt(1/2)gt2 (These equations describe the x and y positions of a projectile that starts at the origin.) You should obtain an equation of the form y=ax+bx2 where a and b are constants.
Two birds begin next to each other and then fly through the air at the same elevation above level ground at 22.5 m/s. One flies northeast, and the other flies northwest. After flying for 10.5 s, what is the distance between them? Ignore the curvature of the Earth.