Practice Problems For Test 2 C ircled problems not on test but may be extra credit on test

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1 Practice Problems For Test 2 C ircled problems not on test but may be extra credit on test Velocity-Acceleration Models Constant Acceleration 1. The skid marks made by an automobile indicated that its brakes were fully applied for a distance of 75 m before it came to a stop. The car in question is known to have a constant deceleration of 20 m/s 2 under these conditions. How fast - in km/hr - was the car traveling when the brakes were first applied? 2. Suppose that a car skids 15 m if it is moving at 50 km/hr when the brakes are applied. Assuming that the car has the same constant deceleration, how far will it skid if it is moving at 100 km/hr when the brakes are applied? 3. A stone is dropped from rest at an initial height h above the surface of the earth. Show that the speed with which it strikes the ground is v = 2gh. 4. Suppose a woman has enough spring in her legs to jump (on earth) from the ground to a height of 2.25 feet. If she jumps straight upward with the same initial velocity on the moon - where the surface gravitational acceleration is (approximately) 5.3 ft/s 2 - how high above the surface will she rise? 5. A bomb is dropped from a helicopter hovering at an altitude of 800 feet above the ground. From the ground directly beneath the helicopter, a projectile is fired straight upward toward the bomb, exactly 2 seconds after the bomb is release. With what initial velocity should the projectile be fired, in order to hit the bomb at an altitude of exactly 400 feet? 6. A spacecraft is in free fall toward the surface of the moon at a speed of 1000 mph. Its retrorockets, when fired, provide a constant deceleration of 20,000 mi/h 2. At what height above the lunar surface should the astronauts fire the retrorockets to insure a soft touchdown? (Ignore the moon s gravitational field) 7. A driver involved in an accident claims he was going only 25 mph. When police tested his car, they found that when its brakes were applied at 25 mph, the car skidded only 45 feet before coming to a stop. But the driver s skid marks at the accident scene measured 210 feet. Assuming the same (constant) deceleration, determine the speed he was actually traveling just prior to the accident. Variable Acceleration 1. Consider the vertical motion of a body with mass m near the surface of the earth (constant gravitational force: g) and a force due to air resistance that is proportional to the velocity of the object. Assume we have an initial velocity of v 0. Set up a differential equation and find the terminal speed of the object. ( lim t v(t) =?) 2. Consider the vertical motion of a body with mass m near the surface of the earth (constant gravitational force: g) and a force due to air resistance that is proportional to the square of velocity of the object. Assume we have an initial velocity of v 0. Set up a differential equation and find the terminal speed of the object. ( lim t v(t) =?) 3 A lunar lander is free-falling toward the moon, and at an altitude of 53 kilometers above the lunar surface its downward velocity is measured at 1477 km/hr. Its retrorockets, when fired in free space, provide a deceleration of T = 4 m/s 2. At what height above the lunar surface should the retrorockets be activated to ensure a soft touchdown? Use Newton s universal law of gravitation F = GMm r 2 G = Nm 2 /kg 2 M = mass of earth or moon where m = mass of object r = distance between centers of the objects Mass of Moon M = kg Radius of Moon R = m 4. What initial velocity is necessary for a projectile to escape from the earth s gravity? Use Newton s law of gravitation. 1

2 Differential Equations - Homework For Test An object with mass m is released from rest at a distance of h meters above the surface of the earth. Use Newton s universal law of gravitation to show that the object impacts the earth surface with a velocity determined by 2ghR v = h + R where g is the acceleration due to gravity at the earth s surface and R is the radius of the earth. Ignore any effects due to earth s rotation or atmosphere. 6. An train engine, starting from rest, pulls a train of mass m using constant power P. The forces resisting to the movement of the train is P dv kv, therefore, the force equation is m v dt = P kv, v(0) = 0. v Explain why the maximum speed of the train is given by P v max = k. Solve the initial value problem, writing the solution in the form t = t(v). The constant k should not be in the final answer. 7. The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity of this sports car. If this machine can accelerate from rest to 100 km/h in ten seconds, how long will it take for the car to accelerate from rest to 200 km/h? 8. Suppose that a motorboat is moving at 40 ft/s when its motor suddenly quits, and that ten seconds later the boat has slowed to 20 ft/s. Assume that the resistance it encounters while coasting is proportional to its velocity. How far will the boat coast in all? 9. Assuming resistance proportional to the square of the velocity, how far does the motorboat of the previous problem coast in the first minute after its motor quits? 10. Suppose that a car starts from rest, its engine providing an acceleration of 10 ft/s 2, while air resistance provide 0.1 ft/s 2 of deceleration for each foot per second of the car s velocity. Find the car s maximum possible velocity. Find how long it takes the car to attain 90% of its limiting velocity, and how far it travels while doing so. 11. A motorboat starts from rest. Its motor provides a constant acceleration of 4 ft/s 2, but water resistance causes a deceleration of v2 400 ft/s 2. Find the velocity after ten seconds. Also find the velocity as t (the maximum possible speed of the boat). 12. An arrow is shot straight upward from the ground with an initial velocity of 160 ft/s. It experiences both the deceleration of gravity and deceleration 800 v2 due to air resistance. How high in the air does it go? Mechanical Vibrations Derivations 1 A mass m is attached to the lower end of a spring that is suspended vertically from a fixed support. Assume we have a frictional force, an external force, and gravity acting on the system. Use Hooke s law and Newton s 2nd law to derive the differential equation that represents the motion of the mass from its static equilibrium position. 2. A mass m is attached to a massless rod of length L. Let θ(t) be the angle in radians with θ = 0 pointing downward. Assume there is no friction for the system. Derive the differential equation that represents the motion of this simple pendulum. Use the law of the conservation of mechanical energy which the sum of the kinetic energy and the potential energy of m remains constant. The potential energy is the product of its weight and its vertical height while the kinetic energy is given by the formula: KE = 1 2 mv2. Linearize the differential equation from part a Free Undamped Motion 1. Assume we have a mass m on a spring with neither damping nor external force. Our differential equation would be mx + kx = 0. Solve this differential equation and manipulate the equation to get the form x(t) = C cos(ω 0 t α) where C = A 2 + B 2 and cos α = C A sin α = C B. What is the amplitude, circular frequency, phase angle, frequency, period, and time lag. (This is an example of simple harmonic motion)

3 Differential Equations - Homework For Test A mass of 3 kg is attached to the end of a spring that is stretched 20 cm by a force of 15 N. It is set in motion with initial position x 0 = 0 and initial velocity v 0 = 10 m/s. Use Hooke s law to set up the differential equation and find the amplitude, period, and frequency of the resulting motion. 3. A body with mass m = 2 1 kg is attached to the end of a spring that is stretched 2 meters by a force of 100 newtons. It is set in motion with initial position x 0 = 1 m and initial velocity v 0 = 5 m/s. Find the position function of the body as well as the amplitude, frequency, period of oscillation, and time lag of its motion. 4. A box with mass 2 kg. stretches a spring 3 cm. If the box is initially at rest, 6 cm above the equilibrium point, find all the times that the box reaches the highest point above the equilibrium position and find the displacement from equilibrium at those times. 5. A box with weight 3 pounds stretches a spring 4 inches. If the box is initially at rest, 1 foot below the equilibrium point, find all the times that the box passes through the equilibrium position. 6. Suppose that in the previous problem, the mass of the box is doubled. What happens to the times when the box passes through the equilibrium position? 7 Consider a floating cylindrical buoy with radius r, height h, and uniform density ρ = 0.5gm/cm 3.The buoy is initially suspended at rest with its bottom at the top surface of the water and is released at time t = 0. Thereafter, it is acted on by two forces: a downward gravitation force equal to its weight mg = ρπr 2 hg and an upward force of buoyancy equal to the weight πr 2 xg of water displaced, where x = x(t) is the depth of the bottom of the buoy beneath the surface of the water at time t. Conclude that the buoy undergoes simple harmonic motion around its equilibrium position x e = ρh with period p = 2π ρh/g. Compute p and the amplitude of the motion if h = 200 cm, g = 980 cm/sec 2, and density of water is 1 gm/cm 3 r 8 A cylindrical buoy weighing 100 lb floats in water with its axis vertical. When depressed slightly and released, it oscillates up and down four times every ten seconds. Assume that friction is negligible. Find the radius of the buoy. 9. A pendulum with no friction can be modeled by the equation d 2 θ dt 2 + g L sin θ = 0 θ(t) = angle, in radians, with θ = 0 where pointing downward L = length of the pendulum, in meters g = 9.8 m/sec 2 Since the above equation is non-linear, the differential equation cannot be solved in elementary terms. However, using the Maclaurin series for sin u is given by sin u = u u3 3! + u5 5! +... for small values of θ, we can approximate the above equation by d 2 θ dt 2 + g L θ = 0 Use this equation, along with L = 1, θ(0) = π, and θ (0) = 0.1, to sketch the graph solution. Using the equation d2 θ dt 2 + g sin θ = 0 L = 1, θ(0) = π, and L θ (0) = 0.1, Mathematica produces the the graph shown in the figure. Explain what the graph is telling you and why it is not the same type of behavior as in part h water level x 10 Plot of t versus θ(t) for θ + 9.8θ = 0, θ(0) = π, θ (0) = 0.1

4 Differential Equations - Homework For Test 2 4 (c) If Mathematica was perfectly accurate, what would Mathematica draw as the solution curve to the following? Explain. Free Damped Motion d 2 θ dt 2 + g L sin θ = 0, θ(0) = π, θ (0) = 0 1. Find and graph the position function for the following initial value problems that represent a mass-spring-dashpot system represented by the differential equation: mx + cx + kx = 0 Also describe what type of damping is taking place: overdamped, critically damped, or underdamped. m = 10 c = 9 k = 2 x(0) = 0 x (0) = 5 m = 25 c = 10 k = 226 x(0) = 20 x (0) = 41 Undamped Forced Oscillations 1. For the differential equation mx + kx = F 0 cos ωt Derive the general solution: x(t) = c 1 cos ω 0 t + c 2 sin ω 0 t + F 0/m ω0 2 cos ωt ω2 k Use the substitution: ω 0 = m Use the initial conditions x(0) = x (0) = 0 to derive the particular solution: F x(t) = 0 m(ω0 2 ω2 ) (cos ωt cos ω 0t) (c) Evaluate lim ω ω 0 F 0 m(ω 2 0 ω2 ) (cos ωt cos ω 0t) (d) Use the trigonometric identity 2 sin A sin B = cos(a B) cos(a + B) to rewrite the solution in 2F part b as: x(t) = 0 m(ω0 2 ω2 ) sin 1 2 (ω 0 ω)t sin 1 2 (ω 0 + ω)t 2. Solve the following initial value problem. sin 50t with a slowly varying amplitude from sin 5t. You can think of the solution as x(t) = A(t) sin 50t where A(t) = sin 5t. 3. A mass weighing 100 lb (mass m = slugs) is attached to the end of a spring that is stretched 1 in. by a force of 100 lb. A force F 0 cos ωt acts on the mass. At what frequency (in hertz) will resonance oscillations occur? Neglect damping. Damped Forced Oscillations 1. A 16-pound weight stretches a spring 8 3 feet. Initially, the weight starts from rest 2 feet below the equilibrium position, and the subsequent motion takes place with a damping force numerically equal to 1 2 the instantaneous velocity. The weight is driven by an external force equal to f (t) = 10 cos ωt. For what value(s) of ω does resonance occur? 2. Find and plot both the transient motion x tr (t) and steady periodic oscillations x sp (t) = C cos(ωt α) of a damped mass-and-spring system: with m = 1, c = 2, and k = 26 under the influence of an external force F(t) = 82 cos 4t with x(0) = 6 and x (0) = 0. with m = 1, c = 4, and k = 5 under the influence of an external force F(t) = 10 cos 3t with x(0) = 0 and x (0) = A small mass m is suspended from a string attached to a wheel-spring system, as shown. The mass of the wheel is M and it has radius R. As the mass moves, the wheel rotates through an angle θ. The spring has a constant k. 0.1x x = 50 cos 45t x(0) = x (0) = 0 Graph the solution x(t) = sin 5t sin 50t in Mathematica. This phenomenon is known as beats. We have a rapid oscillation from

5 Differential Equations - Homework For Test 2 5 R The potential energy of the spring is PE = 1 2 kx2. The potential energy of the mass is PE = mgx. The kinetic energy of the mass is KE = 1 2 mv2. The kinetic energy of the rotating wheel is Two brine tanks where tank 1 contains x(t) pounds of salt in 100 gal of brine and tank 2 contains y(t) pounds of salt in 200 gal of brine. The brine in each tank is kept uniform by sitrring, and brine is pumped from each tank to the other at the rates indicated. In addition, fresh water flows into tank 1. k KE = 1 4 MR2 ω 2. x = 0 The relationships between the variables are m dθ x = Rθ dt = ω Find and solve the differential equation that describes the motion of the mass m. First Order Systems Modeling with a System of Differential Equations (c) The system of two masses and three springs: 1. Write the system of differential equations for the following: The system of two masses and two springs with a given external force f (t) acting on the right hand side of mass m 2. (d) Two particles each of mass m are attached to a string under tension T. Assume that the particles oscillate vertically with amplitudes so small that the sines of the angles shown are accurately approximated by their tangents.

6 Differential Equations - Homework For Test 2 6 (e) Three 100 gallon fermentation vats are connected and the mixtures in each tank are kept uniform by stirring. Denote by x i (t) the amount of alcohol in tank T i at time t. Suppose that the mixture circulates between the tanks at the rate of 10 gal/min. (f) A particle of mass m moves in the plane with coordinates (x(t), y(t)) under the influence of a force that is directed toward the origin and has magnitude k/(x 2 + y 2 ) (an inverse-square central force field). Use the substitution r = x 2 + y 2. (g) Suppose that a projectile of mass m moves in a vertical plane in the atmosphere near the surface of the earth under the influence of two forces: a downward gravitational force of magnitude mg, and a resistive force that is directed opposite to the velocity vector v and has magnitude kv 2 where v = v is the speed of the projectile. Solving a System of Differential Equations Elimination Method 1. The system of differential equations dr dt = aj, dj = br where a > 0, b > 0 dt might be used to describe the affections of two lovers, Romeo and Juliet. If positive values of r and j represent the amount of respective love for the other, Solve the general system, with r(0) = 1, j(0) = 0. In terms of the romance, what is the meaning of the differential equations? (c) Sketch the parametric equation x = j(t), y = r(t) for a = 0.3 and b = 0.1, on the interval 0 t 16. (d) In terms of the romance, what is the effect of the negative sign in the equation dr dt = aj? How is this visible in the graph? (e) What is the effect on the romance if the value of b is increased? adapted from Love Affairs and Differential Equations, by Steven Strogatz, Mathematics Magazine, vol. 1, February Tank A initially contains 1000 gallons of pure water and tank B initially contains 1000 gallons of liquid, containing 50 pounds of dissolved acid. An acid solution containing 0.1 lb/gal of acid is entering tank A at 4 gal/min. The well mixed solution leaves tank A at 6 gal/min and enters tank B. An acid solution containing 0.2 lb/gal of acid is entering tank B at 1 gal/min. The well mixed solution leaves Tank B, with 2 gal/min returning to tank A and 5 gal/min leaving the system. Determine the amount of acid in each tank after the system is run for t minutes. Use Mathematica and the Plot command to create a single graph showing the amount of acid in each tank on the time interval 0 t 1 day. 3. Earlier, you derived the system of differential equations: mx = qβy my = qβx for a particle of mass m and electrical charge q under the influence of the uniform magnetic field B = βk. Suppose that the initial conditions are x(0) = r 0, y(0) = 0, x (0) = 0, y (0) = ωr 0 where ω = qβ/m. Show that the trajectory of the particle is a circle of radius r 0. Operational Determinants Method 1. Find the general solution of the system. Also find the natural frequencies of the mass-spring system and describe its natural modes of oscillation. Let m 1 = 4 m 2 = 2 k 1 = 8 k 2 = 4 and k 3 = 0 2. Set up the system of differential equations assuming m = k = 1 and find the natural frequencies of oscillation of the system.

7 Differential Equations - Homework For Test 2 7 Rewrite the solution in the form: y(x) = (c 1 + c 2 )e ax cos bx + i(c 1 c 2 )e ax sin bx Theory Problems For Test 2 1. Derive the differential equation (with respect to velocity) that represents the vertical motion of mass m near the surface of the earth assuming air resistance is proportional to the square of the velocity. 2. Let y 1 (x) and y 2 (x) be two solutions to the homogeneous linear differential equation: y + p(x)y + q(x)y = 0 Show the superposition principle by proving that y = c 1 y 1 + c 2 y 2 is a solution to the differential equation. 3. Prove that if the functions f and g are linearly dependent, then the Wronskian is identically zero. 4. Given the Homogeneous Linear Second Order Differential Equation with Constant Coefficients: ay + by + cy = 0 use the substitution y = e rx to find the characteristic equation and prove the general solution y(x) = c 1 e r 1x + c 2 e r 2x where r 1 and r 2 are distinct roots. 5. Use the method of Reduction of Order (y 2 (x) = u(x)y 1 (x)) to find a second solution to the following: y + 2y + y = 0 and y 1 (x) = e x is a solution (note: you could of found that solution yourself since we have constant coefficients) x 2 y 5xy + 9y = 0 (x > 0) and y 1 (x) = x 3 is a solution. 6. Assuming r is a complex number, prove that the derivative of e rx is re rx. Use Euler s formula in your proof. 7. Given: ay + by + cy = 0 Assume our characteristic equation has roots that are complex conjugates r 1 = a + bi and r 2 = a bi so our solution is y(x) = c 1 e r 1x + c 2 e r 2x where r 1 and r 2 are complex conjugates. Pick values for c 1 and c 2 so that we get the two independent real-valued solutions: y 1 (x) = e ax cos bx y 2 (x) = e ax sin bx 8. The roots of the characteristic equation fo a certain differential equation are 3, -5, 0, 0,0, 0, -5, 2±3i, 2±3i. Write the general solution of this homogeneous differntial equation. 9. Find a particular solution of y 4y = 2e 2x 10. Determine the appropriate form for a particular solution of the fifth-order equation: (D 2) 3 (D 2 + 9)y = x 2 e 2x + x sin 3x Problems For Test 2: Answers Velocity-Acceleration Models Constant Acceleration Sec (31) v m/s km/hr 2. (32) x s = 60 m 3. (35) v s = 2gh 4. (36) y m ft 5. (41) v ft/sec 6. (42) y 0 = 25 mi 7. (44) v 0 54 mph

8 Differential Equations - Homework For Test 2 8 Variable Acceleration Sec (pg 101) terminal speed = mg k mg 2. (pg 103) terminal speed = k 3. (pg 105 ex4) height above the moon 41, 870 m 2GM 4. (pg 107) v 0 > R 2ghR 5. ( - ) v = h + R 6. ( - ) t = mv2 max 2P ln v 2 max v 2 max v 2 7. (1) t 31.5 sec 8. (3) 577 ft 9. (5) 778 ft 10. (7) 100 ft/sec t sec ft 11. (19) v(10) = 30.5 ft/sec limiting velocity is 40 ft/sec 12. (20) ft Mechanical Vibrations Sec 3.4 Derivations 1. (pg 185) Derivation done in class 2. (pg 186) Derivation done in class Derivation done in class Free Undamped Motion 1. (pg 187) Derivation done in class 2. (3) x(t) = 2 sin 5t amp=2, circ freq = 5, period = 2π/5, freq = 5/2π 5 3. (ex 1) x(t) = 2 cos(10t ) amp = 5/2, circ freq = 10, period = π/5, freq = 5/π, time lag = ( - ) x(t) = 0.06 cos 18.07t t = 2πk (k Z) displacement = ( - ) x(t) = cos 4 6t t = π π 4 k (k Z) 6 6. ( - ) t = π π 4 3 k (k Z) 2 times slower than question five 7. (10) x(t) = ρh(1 cos ω 0 t) equilibrium position = ρh, circ freq = ω 0, period = 2π ρh/g, amp = 100 cm, period = (11) r = in 9. ( - ) Mathematica Problem (c) Free Damped Motion 1. (13) x(t) = 50(e 2 5 t e 1 2 t ) Over-damped (14) x(t) = 25e 1 5 t cos(3t.6435) Under-damped Undamped Forced Oscillations Sec (pg 213) (c)

9 Differential Equations - Homework For Test 2 9 (d) 2. (pg 215 ex2) 3. (19) Damped Forced Oscillations Sec ( - ) 2. (pg 220 ex6) (11) 3. (pg 217) First Order Systems Modeling with a System of Differential Equations Sec (pg 247 ex1) (ex2) (c) (24) (d) (25) (e) (26) (f) (29) (g) (30) Solving a System of Differential Equations Sec 4.2 Elimination Method 1. ( - ) (c) (d) (e) 2. (like ex2 sec 4.1) 3. (35) Operational Determinants Method 1. (39) 2. (47) Theory Problems For Test 2 1. (sec 2.3 pg103) 2. (sec 3.1 pg149) 3. (sec 3.1 pg154) 4. (sec 3.3 pg174) 5. (sec 3.2) (pg 172) (37) 6. (sec 3.3 pg179) 7. (sec 3.3 pg179) 8. (sec 3.3 ex8) 9. (sec 3.5 ex4) 10. (sec 3.5 ex10)