
The path of minimum time to fix the point.

Answer to Problem 6.15P
The path, here, the point 3 is the lowest point on the curve for the path of the roller coaster car.
Explanation of Solution
Figure1
The curve for the path of the roller coaster car between the point 1 and 2
Assume that the car is launched at any height
At point1
Initial kinetic energy of the car is,
Here,
Initial potential energy of the car is,
Here,
The total initial mechanical energy of the car is the sum of the initial kinetic energy of the car and the initial potential energy of the car. That is
Substitute
At any point between 1 and 2, consider the car is moving with a velocity of
According to the law of conservation of energy, the total initial mechanical energy of the car is equal to the total final mechanical energy of the car so,
The expression for unknown path is
The differential form of the variable
Substitute the equation
The expression for the time taken to reach the car from 1 to 2 is,
Substitute
Compare this equation with standard form of Euler-Lagrange equation
Write the Euler-Lagrange equation,
Calculate the value of
Thus, the Euler-Lagrange equation becomes as,
Substitute
But the car raises the height from
Substitute
Take the constant as
Square the equation
Conclusion:
Let
Thus, re-write the equation for
I
Integrate the above expression (V) by substituting ,
Assume that the initial point represent the origin of the path of the car, that is
Substitute
From the properties of the cosine function, the value of the cosine function is equal to 1 for
Substitute
Thus, the expression
Therefore, the final parametric equation for the path of the car is
Here, the point 3 is the lowest point on the curve for the path of the roller coaster car.
Want to see more full solutions like this?
Chapter 6 Solutions
Classical Mechanics
- If you connect a 1.8 F and a 2.6 F capacitor in series, what will be the equivalent capacitance?arrow_forwardSuppose that a particular heart defibrillator uses a 1.5 x 10-5 Farad capacitor. If it is charged up to a voltage of 7300 volts, how much energy is stored in the capacitor? Give your answer as the number of Joules.arrow_forwardThe voltage difference across an 8.3 nanometer thick cell membrane is 6.5 x 10-5volts. What is the magnitude of the electric field inside this cell membrane? (Assume the field is uniform, and give your answer as the number of Volts per meter... which is the same as the number of Newtons per Coulomb.)arrow_forward
- Three identical capacitors are connected in parallel. When this parallel assembly of capacitors is connected to a 12 volt battery, a total of 3.1 x 10-5 coulombs flows through the battery. What is the capacitance of one individual capacitor? (Give your answer as the number of Farads.)arrow_forwardSuppose you construct your own capacitor by placing two parallel plates at a distance 0.27 meters apart. The plates each have a surface area of 0.64 square meters. What is the capacitance of this setup? (Give your answer as the number of Farads.)arrow_forwardDraw a diagram with the new arrows. No they do not point all towards the center.arrow_forward
- Example In Canada, the Earth has B = 0.5 mŢ, pointing north, 70.0° below the horizontal. a) Find the magnetic force on an oxygen ion (O2) moving due east at 250 m/s b) Compare the |FB| to |FE| due to Earth's fair- weather electric field (150 V/m downward).arrow_forwardFour charges, qa, qb, qa, and qd are fixed at the corners of a square. A charge q that is free to move located at the exact center of the square. Classify the scenarios described according to the force that would be exerted on the center charge q. Assume in each case that q is a positive charge. Do not assume that the fixed charges have equal magnitudes unless the scenario defines such an equality. qa Яс q %b Force is zero Force is to the left Force is to the right Force is undeterminedarrow_forwardCharge qi = -q is located at position (0, d). Charge q = −2q₁ is located at position (d,0). Charge q3 = located at position (2d, 2d). 5qi is y Determine the net electric field Ĕ net at the origin. Enter your expression using ij unit vector notation in terms of the given quantities, the permittivity of free space €0, and exact rational and irrational numbers. d 9₁ d TH net = 92 d d Xarrow_forward
- solve pleasearrow_forward= = R4 R5 = 12.5 Q. A - In the circuit shown, R₁ = R₂ = R 3 voltmeter measures the potential difference across the battery. When the switch is in position 1, the voltmeter measures V₁ = 13.8 V. When the switch is in position 2, the voltmeter measures V2 = 13.4 V. What is the emf ☐ of the battery? 14.93 = What is the battery's internal resistance r? r = V CH Ω R₁₂ V S R₁ 02 2 R₁ 4 R3 R 5arrow_forwardConsider the arrangement of charges shown in the figure. Four charges of equal magnitude Q but varying sign are placed at the corners of a square as indicated. A positive charge q is placed in the center. What is the direction of the net force, if any, on the center charge? Indicate your answer by placing the appropriate label in the first box. Then, suppose that the charge q were to be displaced slightly from the center position. On the figure, label each box with the arrow that best indicates the direction of the net force that would act on q if it were moved to that location. Net Force Answer Bank no force ↑ +2 0 -Q -Q +Qarrow_forward
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON





