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Use the following differential equations to compute the velocity and position of a soccer ball that is kicked straight up in the air with an initial velocity of 40 m/s:
where
where
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Chapter 25 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
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Fundamentals of Differential Equations (9th Edition)
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A Graphical Approach to College Algebra (6th Edition)
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- Q: for the following equation, what are the units of x., u, and a if X, is the distance in m. t, is the time in second? [X. x3 = + u?t +arrow_forwardQ1. The image below shows an example of "absolute dependent motion analysis", where the motions of two objects depend on each other, and the goal is to find the relations in their motions (i.e., in their positions, velocities, and accelerations). Please put the suggested steps of analysis in the correct order. VA A B VB Differentiate the entire equation with respect to time, and extract the relation in velocities. Repeat for acceleration if needed. Set up a coordinate (s) along the direction of motion from a fixed point (O) or a fixed datum line. Represent the positions of the objects respectively. In some cases intermediate objects need to be considered too and their positions need to be represented as well. Recognize the constant length(s) and find the depending geometric relations between the position variables.arrow_forwardA cart weighing 0.5 kg is drawn up a smooth 45° incline by a motor, M, winding up a cable. The force in the cable can be expressed as 5t² N, where t is in seconds. When t = 0, the displacement s = 0 and the initial velocity is 3 m/s. Find the cart's velocity when t = 2 seconds. > Draw a very clear FBD of the cart that you can use to write the equations of motion > Write the equations governing the cart's motion along the incline. Use axes parallel and perpendicular to the incline. Find the velocity requestedarrow_forward
- 2- find the center of mass, the velocity of the center of mass, the momentum, and the kinetic energy of the following system: do f(xd) xaf] m₁ = 1 kg T₁=1+2j+3 k v₁ = 2î+3ĵ m₂ = 1 kg T₂ = 1-j+ k v₂ = 2) + 3karrow_forward2) A single degree of freedom mechanical system is provided in the figure, with coordinate markings. Note that all the cables are inflexible, and I, stands for the total inertia of the concentric pulleys. y m 2m mo 3r 2k k momo ||| equivalent system: keq meq Ceq x b) Express the coordinates y and 0 as a function of the generalized coordinate, x. d) Following an energy equivalence approach, and using the coordinate transformations obtained in part (a), determine the equivalent mass, the equivalent spring stiffness, and the equivalent damping coefficient. e) Express the equation of motion of the system, using x, the downward displacement of the block (2m) from the system's equilibrium position as the generalized coordinate. (Use symbolic expressions and show your work, do NOT generate the response).arrow_forwardLook at the below system. Using either the conservation of energy method or Lagrange's method, solve for the governing equation of motion for the system. Put a box around your final answer. Also, put a box around your equations for the potential and kinetic energy of the system. Assume the system's springs are initially unstretched (i.e., assume that there is no gravity until t = 0 [s]). K₁ Î E K₂ Xarrow_forward
- 1. For the following concentration expressions, indicate whether they are uniform or nonuniform and in how many dimensions (OD, 1D, 2D, or 3D), and steady or unsteady. Then for the following control volume and origin, and table of constants, use Excel or Matlab to graph profiles that show how concentration changes within the control volume and over time to a limit of 20 for the following: C(x,0,0,0), C(0,y,0,0), C(0,0,z,0) and C(0,0,0,t). On each graph, show which parameters are held constant, the CV boundaries, and the point where all four plots overlap. 20 C(x=0) 10 a 0.0001 b 0.001 | 20 0.01 k 0.1 100 All of the following functions are C(space, time) and so not necessarily just x as suggested. a. C,(x)= C,(x = 0)x exp{- ax}arrow_forwardYou walk along the beach towards a dock while your friend rows a boat towards the same dock on a flat lake. Your friend's boat approaches the dock on a straight course, but also rotates about its center-of-mass since your friend is not pulling evenly on the oars. If you knew your own velocity (vwalker), the magnitude of boat's angular velocity (thetaboat), and radius vector from yourself to your friend in the boat (rfriend) at any given time, you could use the following equation to calculate the your friend's velocity: vfriend = vwalker + (thetaboat)k x rfriend where k is a unit vector in the vertical direction. True Falsearrow_forward@AB = 4 rad/s %AB = 6 rad/s² 0.5 m 45° B/C 1 m 60° IC Now let's look closer at Figure 2 from the prior question. By drawing lines that pass through points B and C which are perpendicular to their velocities (the red lines in the figure above) we can locate the IC at the intersection (the star). Now we can use some basic trig to figure out the distance TB/Ic labeled in the figure, which is the radial distance from the IC to point B. I've drawn in a green dashed line that creates two right triangles to help with the trig. What is the distance TB/IC? Give your answer in m but don't type the unit.arrow_forward
- . I am planning to perform some volume-flow rate measurements in the Fluid Mechanics Laboratory. For this, I need a volumetric measuring tank (graduated cylinder) and a stopwatch. I considered the volume of the measured tank as 15 gallons and a stopwatch with reaction time as 1/10th of a second (though resolution of 1/1000th of a second). What is the volume flow rate if it takes 5 minutes to fill a 15-gallon of tank? Determine the smallest division to be on the tank in order to estimate the volume flow rate within an accuracy of ± 0.05 gpm.arrow_forwardPhysics 121 Spring 2021 - Document #11: Homework #04 & Reading Assignment page 4 of 8 Problem 1: Gnome Ride - This from a Previous Exam I. A Gnome of given mass M goes on the Gnome Ride as follows: He stands on a horizontal platform that is connected to a large piston so that the platform is driven vertically with a position as a function of time according to the following equation: y(t) = C cos(wt) Here w is a constant given angular frequency, C is a given constant (with appropriate physical units) and y represents the vertical position, positive upward as indicated. Part (a) - What is the velocity of the Gnome at time t = 0? Explain your work. Present your answer in terms of the given parameters Part (b) – What is the net force on the Gnome at time t = 0? Explain your work. Present your answer in terms of the given parameters Part (c) – What is the Normal Force on the Gnome at time t = 0? Explain your work. Present your answer in terms of the given parameters Some Possibly Useful…arrow_forwardA mechanic changing a tire rolls a wheel along the ground towards the car. The radius of the wheel is 42cm, and the speed of the wheel as it rolls is 2 revolutions per second. Height Above Ground (m) radiu HIDE wheel spet Time The diagram above illustrates the vertical motion of a point on the tire over time. It is possible to model the height of this point using a sinusoidal function of the form h(t)=-a sin[b(t-c)]+d. a) Determine the length of time required for one revolution of the tire. b) State the numerical value for each of the parameters a, b, c & d. And write a function representing the motion of the point in the form h(t)= -a sin[b(t−c)]+d.arrow_forward
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