1, 2, 3 = straightforward, intermediate, challenging
Section 4.1 The Position, Velocity, and Acceleration Vectors
1.A motorist drives south at 20.0 m/s for 3.00 min, then turns west and travels at 25.0 m/s for 2.00 min, and finally travels northwest at 30.0 m/s for 1.00 min. For this 6.00-min trip, find (a) the total vector displacement, (b) the average speed, and (c) the average velocity. Let the positive x axis point east.
2.A golf ball is hit off a tee at the edge of a cliff. Its x and y coordinates as functions of time are given by the following expressions:
x = (18.0 m/s)t
and y = (4.00 m/s)t – (4.90 m/s2)t2
(a) Write a vector expression for the ball’s position as a function of time, using the unit vectors [pic] and [pic]. By taking derivatives, obtain expressions for (b) the velocity vector v as a function of time and (c) the acceleration vector a as a function of time. Next use unit-vector notation to write expressions for (d) the position, (e) the velocity, and (f) the acceleration of the golf ball, all at t = 3.00 s.
3.When the Sun is directly overhead, a hawk dives toward the ground with a constant velocity of 5.00 m/s at 60.0( below the horizontal. Calculate the speed of her shadow on the level ground.
4.The coordinates of an object moving in the xy plane vary with time according to the equations x = –(5.00 m) sin(wt) and y = (4.00 m) – (5.00 m)cos(wt), where w is a constant and t is in seconds. (a) Determine the components of velocity and components of acceleration at t = 0. (b) Write expressions for the position vector, the velocity vector, and the acceleration vector at any time t > 0. (c) Describe the path of the object in an xy plot.
Section 4.2 Two-Dimensional Motion with Constant Acceleration
5.At t = 0, a particle moving in the xy plane with constant acceleration has a velocity of [pic] and is at the origin. At t = 3.00 s, the particle's velocity is [pic]. Find (a) the acceleration of the particle and (b) its coordinates at any time t.
6.The vector position of a particle varies in time according to the expression [pic]. (a) Find expressions for the velocity and acceleration as functions of time. (b) Determine the particle's position and velocity at t = 1.00 s.
7.A fish swimming in a horizontal plane has velocity [pic] at a point in the ocean where the position relative to a certain rock is [pic]. After the fish swims with constant acceleration for 20.0 s, its velocity is [pic]. (a) What are the components of the acceleration? (b) What is the direction of the acceleration with respect to unit vector [pic]? (c) If the fish maintains constant acceleration, where is it at t = 25.0 s, and in what direction is it moving?
8.A particle initially located at the origin has an acceleration of [pic]and an initial velocity of [pic]. Find (a) the vector position and velocity at any time t and (b) the coordinates and speed of the particle at t = 2.00 s.
9.It is not possible to see very small objects, such as viruses, using an ordinary light microscope. An electron microscope can view such objects using an electron beam instead of a light beam. Electron microscopy has proved invaluable for investigations of viruses, cell membranes and subcellular structures, bacterial surfaces, visual receptors, chloroplasts, and the contractile properties of muscles. The “lenses” of an electron microscope consist of electric and magnetic fields that control the electron beam. As an example of the manipulation of an electron beam, consider an electron traveling away from the origin along the x axis in the xy plane with initial velocity [pic]. As it passes through the region x = 0 to x = d, the electron experiences acceleration [pic] , where ax and ay are constants. For the case vi = 1.80 ( 107 m/s, ax = 8.00 ( 1014 m/s2 and ay = 1.60 ( 1015m/s2, determine at x = d = 0.0100 m (a) the position of the electron, (b) the velocity of the electron, (c) the...