A projectile that is subject to quadratic air resistance

Then a baseball flies through the air, the ratio fquad/fiin of the quadratic to the linear drag force is given by (2.7). Given that a baseball has diameter 7 cm, find the approximate speed v at which the two drag forces are equally important. For what approximate range of speeds is it safe to treat the drag force as purely quadratic? Under normal conditions is it a good approximation to ignore the linear term? Answer the same questions for a beach ball of diameter 70 cm.

The origin of the linear drag force on a sphere in a fluid is the viscosity of the fluid. According to Stokes's law, the viscous drag on a sphere is flir, = 37r iiDv (2.82) where ri is the viscosity8 of the fluid, D the sphere's diameter, and v its speed. Show that this expression reproduces the form (2.3) for Ain, with b given by (2.4) as b = 8D. Given that the viscosity of air at STP is 17 = 1.7 x 10-5 Ns/m2, verify the value of ,8 given in (2.5).

(a) The quadratic and linear drag forces on a moving sphere in a fluid are given by (2.84) and (2.82) (Problems 2.4 and 2.2). Show that the ratio of these two kinds of drag force can be written as fquad/fiin = R/48,9 where the dimensionless Reynolds number R is R= Dvp (2.83) where D is the sphere's diameter, v its speed, and Q and 77 are the fluid's density and viscosity. Clearly the Reynolds number is a measure of the relative importance of the two kinds of drag.1 When R is 8 For the record, the viscosity of a fluid is defined as follows: Imagine a wide channel along which fluid is flowing (x direction) such that the velocity v is zero at the bottom (y = 0) and increases toward the top (y = h), so that successive layers of fluid slide across one another with a velocity gradient dvldy. The force F with which an area A of any one layer drags the fluid above it is proportional to A and to dv/dy, and )7 is defined as the constant of proportionality; that is, F =ri A dvldy. 9 The numerical factor 48 is for a sphere. A similar result holds for other bodies, but the numerical factor is different for different shapes. 1 The Reynolds number is usually defined by (2.83) for flow involving any object, with D defined as a typical linear dimension. One sometimes hears the claim that R is the ratio f quad,/ du. Since f f quad, lin = R /48 for a sphere, this claim would be better phrased as "R is roughly of the order of fquadf,,f;in." very large, the quadratic drag is dominant and the linear can be neglected; vice versa when R is very small. (b) Find the Reynolds number for a steel ball bearing (diameter 2 mm) moving at 5 cm/s through glycerin (density 1.3 g/cm3 and viscosity 12 Ns/m2 at STP).

The origin of the quadratic drag force on any projectile in a fluid is the inertia of the fluid that the projectile sweeps up. (a) Assuming the projectile has a cross-sectional area A (normal to its velocity) and speed v, and that the density of the fluid is , show that the rate at which the projectile encounters fluid (mass/time) is . (b) Making the simplifying assumption that all of this fluid is accelerated to the speed v of the projectile, show that the net drag force on the projectile is . It is certainly not true that all the fluid that the projectile encounters is accelerated to the full speed v, but one might guess that the actual force would have the form where is a number less than 1, which would depend on the shape of the projectile, with small for a streamlined body, and larger for a body with a flat front end. This proves to be true, and for a sphere the factor is found to be . (c) Show that (2.84) reproduces the form (2.3) for , with given by (2.4) as . Given that the density of air at STP is and that for a sphere, verify the value of given in (2.6).

Suppose that a projectile which is subject to a linear resistive force is thrown vertically down with a speed which is greater than the terminal speed . Describe and explain how the velocity varies with time, and make a plot of against t for the case that .

(a) Equation (2.33) gives the velocity of an object dropped from rest. At first, when is small, air resistance should be unimportant and (2.33) should agree with the elementary result for free fall in a vacuum. Prove that this is the case. [Hint• Remember the Taylor series for , for which the first two or three terms are certainly a good approximation when x is small.] (b) The position of the dropped object is given by (2.35) with . Show similarly that this reduces to the familiar when t is small.

There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity v, that is, . Write down Newton's second law and separate the variables by rewriting it as . Now integrate both sides of this equation and show that Provided you can do the integral, this gives t as a function of v. You can then solve to give v as a function of t. Use this method to solve the special case that , a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and 2.9.

A mass m has velocity at time and coasts along the x axis in a medium where the drag force is . Use the method of Problem 2.7 to find in terms of the time t and the other given parameters. At what time (if any) will it come to rest?

We solved the differential equation (2.29), mi,y = b(vy vmr), for the velocity of an object falling through air, by inspection a most respectable way of solving differential equations. Never-theless, one would sometimes like a more systematic method, and here is one. Rewrite the equation in the "separated" form m dvY b dt V y Vter and integrate both sides from time 0 to t to find vy as a function of t. Compare with (2.30).

For a steel ball bearing (diameter 2 mm and density 7.8 g/cm3) dropped in glycerin (density 1.3 g/cm3 and viscosity 12 Ns/m2 at STP), the dominant drag force is the linear drag given by (2.82) of Problem 2.2. (a) Find the characteristic time t and the terminal speed vt. [In finding the latter, you should include the buoyant force of Archimedes. This just adds a third force on the right side of Equation (2.25).] How long after it is dropped from rest will the ball bearing have reached 95% of its terminal speed? (b) Use (2.82) and (2.84) (with K = 1/4 since the ball bearing is a sphere) to compute the ratioJ quad?

Consider an object that is thrown vertically up with initial speed vo in a linear medium. (a) Measuring y upward from the point of release, write expressions for the object's velocity vy(t) and position y(t). (b) Find the time for the object to reach its highest point and its position ymax at that point. (c) Show that as the drag coefficient approaches zero, your last answer reduces to the well-known result ymax = zvo2/g for an object in the vacuum. [Hint: If the drag force is very small, the terminal speed is very big, so vo/vm, is very small. Use the Taylor series for the log function to approximate ln(1 3) by 3 Z32. (For a little more on Taylor series see Problem 2.18.)]

Problem 2.7 is about a class of one-dimensional problems that can always be reduced to doing an integral. Here is another. Show that if the net force on a one-dimensional particle depends only on position, F = F(x), then Newton's second law can be solved to find v as a function of x given by V2 = V 2 + 2 x f F(x') dx'. 0 x, (2.85) [Hint: Use the chain rule to prove the following handy relation, which we could call the "v dv I dx rule": If you regard v as a function of x, then dv 1 dv2 = v = dx 2 dx (2.86) Use this to rewrite Newton's second law in the separated form m d(v2) = 2F (x) dx and then integrate from xo to x.] Comment on your result for the case that F(x) is actually a constant. (You may recognise your solution as a statement about kinetic energy and work, both of which we shall be discussing in Chapter 4.)

Consider a mass m constrained to move on the x axis and subject to a net force F = kx where k is a positive constant. The mass is released from rest at x = x0 at time t = 0. Use the result (2.85) in Problem 2.12 to find the mass's speed as a function of x; that is, dx I dt = g(x) for some function g(x). Separate this as dx I g(x) = dt and integrate from time 0 to t to find x as a function of t. (You may recognize this as one way not the easiest to solve the simple harmonic oscillator.)

Use the method of Problem 2.7 to solve the following: A mass m is constrained to move along the x axis subject to a force F (v) = Foe" I V, where Fo and V are constants. (a) Find v(t) if the initial velocity is vo > 0 at time t = 0. (b) At what time does it come instantaneously to rest? (c) By integrating v(t), you can find x(t). Do this and find how far the mass travels before coming instantaneously to rest.

Consider a projectile launched with velocity (v,o, vyo) from horizontal ground (with x measured horizontally and y vertically up). Assuming no air resistance, find how long the projectile is in the air and show that the distance it travels before landing (the horizontal range) is 2vxovy/ g.

A golfer hits his ball with speed vo at an angle 6 above the horizontal ground. Assuming that the angle 0 is fixed and that air resistance can be neglected, what is the minimum speed vo(min) for which the ball will clear a wall of height h, a distanced away? Your solution should get into trouble if the angle 0 is such that tan 0 < h 1 d. Explain. What is vo(min) if 6 = 25, d = 50 m, and h = 2 m?

The two equations (2.36) give a projectile's position (x, y) as a function of t. Eliminate t to give y as a function of x. Verify Equation (2.37).

Taylor's theorem states that, for any reasonable function f (x), the value of f at a point (x 6) can be expressed as an infinite series involving f and its derivatives at the point x: f + 8) = f (x) + f'(x)8 + 2 1 ! f (x)82 + 3 1 r(x)63 (2.87) where the primes denote successive derivatives of f (x). (Depending on the function this series may converge for any increment 3 or only for values of 8 less than some nonzero "radius of convergence.") This theorem is enormously useful, especially for small values of 3, when the first one or two terms of the series are often an excellent approximation." (a) Find the Taylor series for ln(1 + 8). (b) Do the same for cos 8. (c) Likewise sin 3. (d) And e8.

Consider the projectile of Section 2.3. (a) Assuming there is no air resistance, write down the position (x, y) as a function of t, and eliminate t to give the trajectory y as a function of x. (b) The correct trajectory, including a linear drag force, is given by (2.37). Show that this reduces to your answer for part (a) when air resistance is switched off (r and vt, = gr both approach infinity). [Hint: Remember the Taylor series (2.40) for ln(1 - ).]

[Computer] Use suitable graph-plotting software to plot graphs of the trajectory (2.36) of a projectile thrown at 45above the horizontal and subject to linear air resistance for four different values of the drag coefficient, ranging from a significant amount of drag down to no drag at all. Put all four trajectories on the same plot. [Hint: In the absence of any given numbers, you may as well choose convenient values. For example, why not take vxo = vyo = 1 and g = 1. (This amounts to choosing your units of length and time so that these parameters have the value 1.) With these choices, the strength of the drag is given by the one parameter vt = t, and you might choose to plot the trajectories for Vter = 0-351, 3, and oo (that is, no drag at all), and for times from t = 0 to 3. For the case that vter = oo, you'll probably want to write out the trajectory separately.]

A gun can fire shells in any direction with the same speed vo. Ignoring air resistance and using cylindrical polar coordinates with the gun at the origin and z measured vertically up, show that the gun can hit any object inside the surface V 2 g 2 Z = 2 2v p 0 Describe this surface and comment on its dimensions.

[Computer] The equation (2.39) for the range of a projectile in a linear medium cannot be solved analytically in terms of elementary functions. If you put in numbers for the several parameters, then it can be solved numerically using any of several software packages such as Mathematica, Maple, and MatLab. To practice this, do the following: Consider a projectile launched at angle 0 above the horizontal ground with initial speed v0 in a linear medium. Choose units such that vo = 1 and g = 1. Suppose also that the terminal speed vter = 1. (With vo = vter, air resistance should be fairly important.) We know that in a vacuum, the maximum range occurs at 0 = r/4 0.75. (a) What is the maximum range in a vacuum? (b) Now solve (2.39) for the range in the given medium at the same angle 0 = 0.75. (c) Once you have your calculation working, repeat it for some selection of values of 0 within which the maximum range probably lies. (You could try 0 = 0.4, 0.5, , 0.8.) (d) Based on these results, choose a smaller interval for 0 where you're sure the maximum lies and repeat the process. Repeat it again if necessary until you know the maximum range and the corresponding angle to two significant figures. Compare with the vacuum values.

Find the terminal speeds in air of (a) a steel ball bearing of diameter 3 mm, (b) a 16-pound steel shot, and (c) a 200-pound parachutist in free fall in the fetal position. In all three cases, you can safely assume the drag force is purely quadratic. The density of steel is about 8 g/cm3 and you can treat the parachutist as a sphere of density 1 g/cm3.

Consider a sphere (diameter D, density psph) falling through air (density Qair) and assume that the drag force is purely quadratic. (a) Use Equation (2.84) from Problem 2.4 (with K = 1/4 for a sphere) to show that the terminal speed is vter = 8 D g Osph 3 Qair (2.88) (b) Use this result to show that of two spheres of the same size, the denser one will eventually fall faster. (c) For two spheres of the same material, show that the larger will eventually fall faster.

Consider the cyclist of Section 2.4, coasting to a halt under the influence of a quadratic drag force. Derive in detail the results (2.49) and (2.51) for her velocity and position, and verify that the constant r = m/cvo is indeed a time.

A typical value for the coefficient of quadratic air resistance on a cyclist is around c = 0.20 N/(m/s)2. Assuming that the total mass (cyclist plus cycle) is m = 80 kg and that at t = 0 the cyclist has an initial speed vo = 20 m/s (about 45 mi/h) and starts to coast to a stop under the influence of air resistance, find the characteristic time r = m/cvo. How long will it take him to slow to 15 m/s? What about 10 m/s? And 5 m/s? (Below about 5 m/s, it is certainly not reasonable to ignore friction, so there is no point pursuing this calculation to lower speeds.)

I kick a puck of mass m up an incline (angle of slope = 0) with initial speed vo. There is no friction between the puck and the incline, but there is air resistance with magnitude f (v) = cv2. Write down and solve Newton's second law for the puck's velocity as a function oft on the upward journey. How long does the upward journey last?

A mass m has speed vo at the origin and coasts along the x axis in a medium where the drag force is F(v) = cv312. Use the "v dv/dx rule" (2.86) in Problem 2.12 to write the equation of motion in the separated form m v dvl F (v) = dx, and then integrate both sides to give x in terms of v (or vice versa). Show that it will eventually travel a distance 2m lvo/c.

The terminal speed of a 70-kg skydiver in spread-eagle position is around 50 m/s (about 115 mi/h). Find his speed at times t = 1, 5, 10, 20, 30 seconds after he jumps from a stationary balloon. Compare with the corresponding speeds if there were no air resistance.

Suppose we wish to approximate the skydiver of Problem 2.29 as a sphere (not a very promising approximation, but nevertheless the kind of approximation physicists sometimes like to make). Given the mass and terminal speed, what should we use for the diameter of the sphere? Does your answer seem reasonable?

A basketball has mass m = 600 g and diameter D = 24 cm. (a) What is its terminal speed? (b) If it is dropped from a 30-m tower, how long does it take to hit the ground and how fast is it going when it does so? Compare with the corresponding numbers in a vacuum.

Consider the following statement: If at all times during a projectile's flight its speed is much less than the terminal speed, the effects of air resistance are usually very small. (a) Without reference to the explicit equations for the magnitude of vt, explain clearly why this is so. (b) By examining the explicit formulas (2.26) and (2.53) explain why the statement above is even more useful for the case of quadratic drag than for the linear case. [Hint: Express the ratio f 1 mg of the drag to the weight in terms of the ratio v /vter.]

The hyperbolic functions cosh z and sinh z are defined as follows: ez e' cosh z = and sinh z = ez ez 2 2 for any z, real or complex. (a) Sketch the behavior of both functions over a suitable range of real values of z. (b) Show that cosh z = cos(iz). What is the corresponding relation for sinh z? (c) What are the derivatives of cosh z and sinh z? What about their integrals? (d) Show that cosh2 z sinh2 z = 1. (e) Show that f dx 1,\/1 x2 = arcsinh x. [Hint: One way to do this is to make the substitution x = sinh z.]

The hyperbolic function tanh z is defined as tanh z = sinh z/ cosh z, with cosh z and sinh z defined as in Problem 2.33. (a) Prove that tanh z = tan(iz). (b) What is the derivative of tanh z? (c) Show that f dz tanh z = In cosh z. (d) Prove that 1 tanh2 z = sech2z, where sech z = 1/ cosh z. (e) Show that f dx/(1 x2) = arctanh x .

(a) Fill in the details of the arguments leading from the equation of motion (2.52) to Equations (2.57) and (2.58) for the velocity and position of a dropped object subject to quadratic air resistance. Be sure to do the two integrals involved. (The results of Problem 2.34 will help.) (b) Tidy the two equations by introducing the parameter r = vtIg Show that when t = r, v has reached 76% of its terminal value. What are the corresponding percentages when t = 2r and 3z? (c) Show that when t >> r, the position is approximately y v t ter const. [Hint: The definition of cosh x (Problem 2.33) gives you a simple approximation when x is large.] (d) Show that for t small, Equation (2.58) for the position gives y ti Igt2. [Use the Taylor series for cosh x and for ln(1 +

Consider the following quote from Galileo's Dialogues Concerning Two New Sciences: Aristotle says that "an iron ball of 100 pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit." I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths. We know that the statement attributed to Aristotle is totally wrong, but just how close is Galileo's claim that the difference is just "two finger breadths"? (a) Given that the density of iron is about 8 g/cm3, find the terminal speeds of the two iron balls. (b) Given that a cubit is about 2 feet, use Equation (2.58) to find the time for the heavier ball to land and then the position of the lighter ball at that time. How far apart are they?

The result (2.57) for the velocity of a falling object was found by integrating Equation (2.55) and the quickest way to do this is to use the integral f du I (1 u2) = arctanh u. Here is another way to do it: Integrate (2.55) using the method of "partial fractions," writing 1 _1C1 1 1 u2 - 2 1 + u 1u which lets you do the integral in terms of natural logs. Solve the resulting equation to give v as a function of t and show that your answer agrees with (2.57).

A projectile that is subject to quadratic air resistance is thrown vertically up with initial speed vo. (a) Write down the equation of motion for the upward motion and solve it to give v as a function of t. (b) Show that the time to reach the top of the trajectory is trop = (vterl g) arctan (vo / vt) . (c) For the baseball of Example 2.5 (with titer = 35 m/s), find trop for the cases that vo = 1, 10, 20, 30, and 40 m/s, and compare with the corresponding values in a vacuum.

When a cyclist coasts to a stop, he is actually subject to two forces, the quadratic force of air resistance, f = cv2 (with c as given in Problem 2.26), and a constant frictional force ft., of about 3 N. The former is dominant at high and medium speeds, the latter at low speed. (The frictional force is a combination of ordinary friction in the bearings and rolling friction of the tires on the road.) (a) Write down the equation of motion while the cyclist is coasting to a stop. Solve it by separating variables to give t as a function of v. (b) Using the numbers of Problem 2.26 (and the value ff, = 3 N given above) find how long it takes the cyclist to slow from his initial 20 m/s to 15 m/s. How long to slow to 10 and 5 m/s? How long to come to a full stop? If you did Problem 2.26, compare with the answers you got there ignoring friction entirely.

Consider an object that is coasting horizontally (positive x direction) subject to a drag force f = by cv2. Write down Newton's second law for this object and solve for v by separating variables. Sketch the behavior of v as a function of t. Explain the time dependence for t large. (Which force term is dominant when t is large?)

A baseball is thrown vertically up with speed vo and is subject to a quadratic drag with magnitude f (v) = cv2. Write down the equation of motion for the upward journey (measuring y vertically up) and show that it can be rewritten as i) = g[l + (v/vt)2]. Use the "v dv dx rule" (2.86) to write I) as v dvldy, and then solve the equation of motion by separating variables (put all terms involving v on one side and all terms involving y on the other). Integrate both sides to give y in terms of v, and hence v as a function of y. Show that the baseball's maximum height is 2v vo 2 2 Vter in ter Ymax = 2 2g + V ter (2.89) If vo = 20 m/s (about 45 mph) and the baseball has the parameters given in Example 2.5 (page 61), what is ym? Compare with the value in a vacuum.

Consider again the baseball of Problem 2.41 and write down the equation of motion for the downward journey. (Notice that with a quadratic drag the downward equation is different from the upward one, and has to be treated separately.) Find v as a function of y and, given that the downward journey starts at ym as given in (2.89), show that the speed when the ball returns to the ground is vtvoilvL vo2. Discuss this result for the cases of very much and very little air resistance. What is the numerical value of this speed for the baseball of Problem 2.41? Compare with the value in a vacuum.

[Computer] The basketball of Problem 2.31 is thrown from a height of 2 m with initial velocity vo = 15 m/s at 45 above the horizontal. (a) Use appropriate software to solve the equations of motion (2.61) for the ball's position (x, y) and plot the trajectory. Show the corresponding trajectory in the absence of air resistance. (b) Use your plot to find how far the ball travels in the horizontal direction before it hits the floor. Compare with the corresponding range in a vacuum.

Computer] To get an accurate trajectory for a projectile one must often take account of several complications. For example, if a projectile goes very high then we have to allow for the reduction in air resistance as atmospheric density decreases. To illustrate this, consider an iron cannonball (diameter 15 cm, density 7.8 g/cm3) that is fired with initial velocity 300 m/s at 50 degrees above the horizontal. The drag force is approximately quadratic, but since the drag is proportional to the atmospheric density and the density falls off exponentially with height, the drag force is f = c(y)v2 where c(y) = y D2 exp(y/A) with y given by (2.6) and A ti 10, 000 m. (a) Write down the equations of motion for the cannonball and use appropriate software to solve numerically for x(t) and y(t) for 0 < t < 3.5 s. Plot the ball's trajectory and find its horizontal range. (b) Do the same calculation ignoring the variation of atmospheric density [that is, setting c(y) = c(0)], and yet again ignoring air resistance entirely. Plot all three trajectories for 0 < t < 3.5 s on the same graph. You will find that in this case air resistance makes a huge difference and that the variation of air resistance makes a small, but not negligible, difference

(a) Using Euler's relation (2.76), prove that any complex number z = x iy can be written in the form z = rei , where r and 0 are real. Describe the significance of r and 0 with reference to the complex plane. (b) Write z = 3 + 4i in the form z = reie (c) Write z = 2e-i7113 in the form x iy.

For any complex number z = x iy, the real and imaginary parts are defined as the real numbers Re(z) = x and Im(z) = y. The modulus or absolute value is lz I = \/x2 + y2 and the phase or angle is the value of 6 when z is expressed as z = re . The complex conjugate is z* = x iy. (This last is the notation used by most physicists; most mathematicians use -z-.) For each of the following complex numbers, find the real and imaginary parts, the modulus and phase, and the complex conjugate, and sketch z and z* in the complex plane: (a) z = 1 + i (b) z = 1 (c) z = ,4e-in (d) z 5eiwt. In part (d), w is a constant and t is the time.

For each of the following two pairs of numbers compute z + w, z w, zw, and z/w. (a) z = 6 + 8i and w = 3 4i (b) z = 8eiTh/3 and w = 4ei'/6. Notice that for adding and subtracting complex numbers, the form x + iy is more convenient, but for multiplying and especially dividing, the form rei9 is more convenient. In part (a), a clever trick for finding z/w without converting to the form rei is to multiply top and bottom by w*; try this one both ways.

Prove that lz I = ,N/z*z for any complex number z.

Consider the complex number z = e'9 = cos 8 + i sin 9. (a) By evaluating z2 two different ways, prove the trig identities cos 26 = cos2 9 sin2 9 and sin 29 = 2 sin 8 cos 9. (b) Use the same technique to find corresponding identities for cos 39 and sin 39.

Use the series definition (2.72) of ez to prove that12 dez ldz = ez.

Use the series definition (2.72) of ez to prove that ezew = ez+w. . [Hint: If you write down the left side as a product of two series, you will have a huge sum of terms like zn Wm. If you group together all the terms for which n + m is the same (call it p) and use the binomial theorem, you will find you have the series for the right side.

The transverse velocity of the particle in Sections 2.5 and 2.7 is contained in (2.77), since + ivy. By taking the real and imaginary parts, find expressions for vx and vy separately. Based on these expressions describe the time dependence of the transverse velocity.

A charged particle of mass m and positive charge q moves in uniform electric and magnetic fields, E and B, both pointing in the z direction. The net force on the particle is F = q(E + v x B). Write down the equation of motion for the particle and resolve it into its three components. Solve the equations and describe the particle's motion.

In Section 2.5 we solved the equations of motion (2.68) for the transverse velocity of a charge in a magnetic field by the trick of using the complex number ri = v, + i vy. As you might imagine, the equations can certainly be solved without this trick. Here is one way: (a) Differentiate the first of equations (2.68) with respect to t and use the second to give you a second-order differential equation for vx. This is an equation you should recognize [if not, look at Equation (1.55)] and you can write down its general solution. Once you know vx, (2.68) tells you vy. (b) Show that the general solution you get here is the same as the general solution contained in (2.77), as disentangled in Problem 2.52.

A charged particle of mass m and positive charge q moves in uniform electric and magnetic fields, E pointing in the y direction and B in the z direction (an arrangement called "crossed E and B fields"). Suppose the particle is initially at the origin and is given a kick at time t = 0 along the x axis with vx = vxo (positive or negative). (a) Write down the equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane z = 0. (b) Prove that there is a unique value of vxo, called the drift speed vdr, for which the particle moves undeflected through the fields. (This is the basis of velocity selectors, which select particles traveling at one chosen speed from a beam with many different speeds.) (c) Solve the equations of motion to give the particle's velocity as a function of t, for arbitrary values of vx0. [Hint: The equations for (vx, vy) should look very like Equations (2.68) except for an offset of vx by a constant. If you make a change of variables of the form U., = vd, and uy = vy, the equations for (ux, uy) will have exactly the form (2.68), whose general solution you know.] (d) Integrate the velocity to find the position as a function of t and sketch the trajectory for various values of vx.o.