Math

CalculusMultivariable Calculus(a) A projectile it fired from the origin down an inclined plain: that makes an angle θ with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are α and r v . respectively. Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of the time t . (Ignore air resistance.) (b) Show that the angle of elevation α that will maximize the downhill range it the angle halfway between the plane and the vertical. (c) Suppose the projectile it fired up an inclined plane whose angle of inclination is θ Show that, in order to maximize the (uphill) range, the projectile should be fired in the direction halfway between the plane and the vertical. (d) In a paper presented in 1686. Edmond Hailey summarized the laws of gravity and projectile motion and applied them to gunnery. One problem he posed involved firing a projectile to hit a target a distance 17 up an inclined plane. Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angle in part (c). (Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.) FIGURE FOR PROBLEM 4BuyFind*launch*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266643

Chapter 13, Problem 4P

Textbook Problem

(a) A projectile it fired from the origin down an inclined plain: that makes an angle *θ* with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are *α* and *r _{v}*. respectively. Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of the time

(b) Show that the angle of elevation *α* that will maximize the downhill range it the angle halfway between the plane and the vertical.

(c) Suppose the projectile it fired up an inclined plane whose angle of inclination is *θ* Show that, in order to maximize the (uphill) range, the projectile should be fired in the direction halfway between the plane and the vertical.

(d) In a paper presented in 1686. Edmond Hailey summarized the laws of gravity and projectile motion and applied them to gunnery. One problem he posed involved firing a projectile to hit a target a distance 17 up an inclined plane. Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angle in part (c). (Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.)

**FIGURE FOR PROBLEM 4**

Multivariable Calculus

Ch. 13.1 - 1-2 Find the domain of the vector function. 1....Ch. 13.1 - Find the domain of the vector function. 2....Ch. 13.1 - Find the limit. 3. limt0(e3ti+t2sin2tj+cos2tk)Ch. 13.1 - Find the limit. 4. limt1(t2-tt-1i+t+8j+sintlntk)Ch. 13.1 - Find the limit. 5. limt1+t21t2,tan-1t,1e2ttCh. 13.1 - Find the limit. 6. limttet,t3+12t3-1,tsin1tCh. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....

Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Draw the projections of the curve on the three...Ch. 13.1 - Draw the projections of the curve on the three...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Find three different surfaces that contain the...Ch. 13.1 - Find three different surfaces that contain the...Ch. 13.1 - At what points does the curve r(t) = t i + (2t ...Ch. 13.1 - At what points does the helix r(t) = sin t, cos t,...Ch. 13.1 - Graph the curve with parametric equations x = sin...Ch. 13.1 - Graph the curve with parametric equations x = (1 +...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Two particles travel along the space curves r1 (t)...Ch. 13.1 - Suppose u and v are vector functions that possess...Ch. 13.2 - The figure shows a curve C given by a vector...Ch. 13.2 - (a) Make a large sketch of the curve described by...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - Find the derivative of the vector function. 9....Ch. 13.2 - Find the derivative of the vector function. 10....Ch. 13.2 - Find the derivative of the vector function. 11....Ch. 13.2 - Find the derivative of the vector function. 12....Ch. 13.2 - Find the derivative of the vector function. 13....Ch. 13.2 - Find the derivative of the vector function. 14....Ch. 13.2 - Find the derivative of the vector function. 15....Ch. 13.2 - Find the derivative of the vector function. 16....Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - If r(t) = t, t2, t3, find r'(t), T( 1), r"(t). and...Ch. 13.2 - If r(t) = e2t, e2t, te2t, find T(0), r"(0), and...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find a vector equation for the tangent line to the...Ch. 13.2 - Find the point on the curve r(t) = 2 cos t, 2 sin...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - (a) Find the point of intersection of the tangent...Ch. 13.2 - The curves r1(t) = t, t2, t3 and r2(t) = sin t,...Ch. 13.2 - At what point do the curves r1(t) = t, 1 t, 3 +...Ch. 13.2 - Evaluate the integral. 35. 02(ti-t3j+3t5k)dtCh. 13.2 - Evaluate the integral. 36. 14(2t3/2i+(t+1)tk)dtCh. 13.2 - Evaluate the integral. 37....Ch. 13.2 - Evaluate the integral. 38....Ch. 13.2 - Evaluate the integral. 39....Ch. 13.2 - Evaluate the integral. 40. (te2ti+t1-tj+11-t2k)dtCh. 13.2 - Find r(t) if r'(t) = 2t i + 3t2 j + t k and r(1) =...Ch. 13.2 - Find r(t) if r'(t) = t i + et j + tet k and r(0) =...Ch. 13.2 - Prove Formula 1 of Theorem 3.Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prove Formula 5 of Theorem 3.Ch. 13.2 - Prove Formula 6 of Theorem 3.Ch. 13.2 - If u(t) = sin t, cos t, t) and v(t) = t, cos t,...Ch. 13.2 - If u and v are the vector functions in Exercise...Ch. 13.2 - Find f'(2), where f(t) = u(t) v(t), u(2) = 1, 2,...Ch. 13.2 - If r(t) = u(t) v(t), where u and v are the vector...Ch. 13.2 - If r(t) = a cos t + b sin t, where a and b are...Ch. 13.2 - If r is the vector function in Exercise 51, show...Ch. 13.2 - Show that if r is a vector function such that r''...Ch. 13.2 - Find an expression for ddt[u(t)(v(t)w(t))].Ch. 13.2 - If r(t) 0, show that ddtr(t)=1r(t)r(t)r(t)....Ch. 13.2 - If a curve has the property that the position...Ch. 13.2 - If u(t) = r(t)[r'(t) r''(t)], show that u(t) =...Ch. 13.2 - Show that the tangent vector to a curve defined by...Ch. 13.3 - Find the length of the curve. 1. r(t) =t, 3 cos t,...Ch. 13.3 - Find the length of the curve. 2. r(t)=2t,t2,13t3,...Ch. 13.3 - Find the length of the curve. 3. r(t)=2ti+etj+etk,...Ch. 13.3 - Find the length of the curve. 4. r(t) =cos t i +...Ch. 13.3 - Find the length of the curve. 5. r(t) = i + t2 j +...Ch. 13.3 - Find the length of the curve. 6. r(t) = t2 i + 9t...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Graph the curve with parametric equations x = sin...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Find, correct to four decimal places, the length...Ch. 13.3 - (a) Find the arc length function for the curve...Ch. 13.3 - (a) Find the arc length function for the curve...Ch. 13.3 - Suppose you start at the point (0, 0. 3) and move...Ch. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - Use Theorem 10 to find the curvature. 21. r(t) =...Ch. 13.3 - Use Theorem 10 to find the curvature. 22. r(t) = t...Ch. 13.3 - Use Theorem 10 to find the curvature. 23....Ch. 13.3 - Find the curvature of r(t) = t2, ln t, t ln t at...Ch. 13.3 - Find the curvature of r(t) = t, t2, t3 at the...Ch. 13.3 - Graph the curve with parametric equations x = cos...Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - To find: The curvature of y=tanx using Formula 11....Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - (a) Is the curvature of the curve C shown in the...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve y =...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve y =...Ch. 13.3 - Use Theorem 10 to show that the curvature of a...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Consider the curvature at x = 0 for each member of...Ch. 13.3 - Find the vectors T, N, and B at the given point....Ch. 13.3 - Find the vectors T, N, and B at the given point....Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - At what point on the curve x = t3, y = 3t, z = t4...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Show that the osculating plane at every point on...Ch. 13.3 - The rectifying plane of a curve at a point is the...Ch. 13.3 - Show that the curvature is related to the tangent...Ch. 13.3 - Show that the curvature of a plane curve is =...Ch. 13.3 - To deduce: the Formula dNds=KT+B. Solution: From...Ch. 13.3 - Use ihe Frenet-Serret formulas to prove each of...Ch. 13.3 - Show that the circular helix r(t) = a cos t, a sin...Ch. 13.3 - Use the formula in Exercise 63(d) to find the...Ch. 13.3 - Find the curvature and torsion of the curve x =...Ch. 13.3 - The DNA molecule has the shape of a double helix...Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - The position function of a particle is given by...Ch. 13.4 - What force is required so that a particle of mass...Ch. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Rework Exercise 23 if the projectile is fired from...Ch. 13.4 - A ball is thrown at an angle of 45 to the ground....Ch. 13.4 - A projectile is tired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - A batter hits a baseball 3 ft above the ground...Ch. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Show that a projectile reaches three-quarters of...Ch. 13.4 - A ball is thrown eastward into the air from the...Ch. 13.4 - A ball with mass 0.8 kg is thrown southward into...Ch. 13.4 - Another reasonable model for the water speed of...Ch. 13.4 - A particle has position function r(t). If r(t) = c...Ch. 13.4 - (a) If a particle moves along a straight line,...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - If a particle with mass m moves with position...Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - A rocket burning its onboard fuel while moving...Ch. 13 - What is a vector function? How do you find its...Ch. 13 - What is the connection between vector functions...Ch. 13 - How do you find the tangent vector to a smooth...Ch. 13 - If u and v are differentiable vector functions, c...Ch. 13 - How do you find the length of a space curve given...Ch. 13 - (a) What is the definition of curvature? (b) Write...Ch. 13 - (a) Write formulas for the unit normal and...Ch. 13 - (a) How do you find the velocity, speed, and...Ch. 13 - State Keplers Laws.Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - (a) Sketch the curve with vector function r(t) = t...Ch. 13 - Let r(t) = 2-t, (et 1)/t, ln(t + 1). (a) Find the...Ch. 13 - Find a vector function that represents the curve...Ch. 13 - Find parametric equations for the tangent line to...Ch. 13 - If r(t) = t2 i + t cos t j + sin t k, evaluate...Ch. 13 - Let C be the curve with equations x = 2 t3 y = 2t...Ch. 13 - Use Simpsons Rule with n = 6 to estimate the...Ch. 13 - Find the length of the curve r(t) = 2t3/2, cos 2t,...Ch. 13 - The helix r1(t) = cos t i + sin t j + t k...Ch. 13 - Reparametrize the curve r(t) = et i + et sin t j +...Ch. 13 - For the curve given by r(t) = sin3 t, cos3 t, sin2...Ch. 13 - Find the curvature of the ellipse x = 3 cos t, y =...Ch. 13 - Find the curvature of the curve y = x4 at the...Ch. 13 - Find an equation of the osculating circle of the...Ch. 13 - Find an equation of the osculating plane of the...Ch. 13 - The figure shows the curve C traced by a particle...Ch. 13 - A particle moves with position function r(t) = t...Ch. 13 - Find the velocity, speed, and acceleration of a...Ch. 13 - A particle starts at the origin with initial...Ch. 13 - An athlete throws a shot at an angle of 45 to the...Ch. 13 - A projectile is launched with an initial speed of...Ch. 13 - Find the tangential and normal components of the...Ch. 13 - A disk of radius 1 is rotating in the...Ch. 13 - PROBLEM PLUS FIGURE FOR PROBLEM 1 1. A particle P...Ch. 13 - A circular curve of radius R on a highway is...Ch. 13 - A projectile is fired from the origin with angle...Ch. 13 - (a) A projectile it fired from the origin down an...Ch. 13 - A ball rolls off a table with a speed of 2 ft/s....Ch. 13 - Find the curvature of the curve with parametric...Ch. 13 - If a projectile is fired with angle of elevation ...Ch. 13 - A cable has radius r and length L and is wound...Ch. 13 - Show that the curve with vector equation r(t) =...

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