# The Hitchhiker's Guide to Calculus

by C. Bond The following sections include solutions to a number of my favorite problems in elementary physics. Some of the solutions bear aspects resembling that of a magician pulling a rabbit out of a hat. Others simply demonstrate the remarkable power of a few seminal concepts to reveal the inner workings of the real world. Most of the problems yield to solution strategies other than the ones shown, but these represent my own preference. At some point in time, I expect to post similar documents containing problems of a more advanced nature, but the problems here may interest physicists and students at all levels.

Contents

1 Kinematics Equations 1.1 Miscellaneous Problems in Kinematics . . . . . . . . . . . . . 1.1.1 1.1.2 1.1.3 Minimum Time for a Vehicle to go from 0 to 60 mph. Minimum Stopping Distance . . . . . . . . . . . . . . Flight of the Bumblebee . . . . . . . . . . . . . . . . . 5 7 7 7 8 8 11

2 Bouncing Ball 3 Maximum Velocity in a Quarter Mile 1

4 Rolling Up A Ramp 4.1 4.2 Maximum Height of Ball . . . . . . . . . . . . . . . . . . . . . Hoop, Disk, Cylinder and Sphere . . . . . . . . . . . . . . . .

11 11 13 14 15 17 18 18 19 20 22 24 24 26 27 28 28 30

5 Height of Water in Tank 6 Bead Sliding on Wire 7 James Bond’s Ski Saga 8 Moment of Inertia 8.1 8.2 8.3 8.4 Constant Moment Arm . . . . . . . . . . . . . . . . . . . . . . Moment of Disk or Solid Cylinder About Axis . . . . . . . . Moment of Thin Spherical Shell About Axis . . . . . . . . . . Moment of Solid Sphere About Axis . . . . . . . . . . . . . .

9 Vertical Loop 9.1 9.2 9.3 Ball on String . . . . . . . . . . . . . . . . . . . . . . . . . . . Cart on Track . . . . . . . . . . . . . . . . . . . . . . . . . . . Pole Vault . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Cue Ball Slip Problems 10.1 Slip Problem #1 . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Slip Problem #2 . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Object on a Bowling Ball 11.1 Bug on a Bowling Ball . . . . . . . . . . . . . . . . . . . . . . 11.2 Marble on a Bowling Ball . . . . . . . . . . . . . . . . . . . . . 12 Volume of Solid Ring 13 Orbital Velocity for Low Earth Orbit 14 Escape Velocity 15 Geosynchronous Orbit 16 Simple Harmonic Motion 17 Gravitational Field Inside a Spherical Shell 18 Tunnel Through the Earth 19 Snell’s Law 20 Mirrors and Lenses 20.1 Finding the Focal Point . . . . . . . . . . . . . . . . . . . . . . 20.2 The Mirror/Lens Equation . . . . . . . . . . . . . . . . . . . . 20.3 Lensmaker’s Formula . . . . . . . . . . . . . . . . . . . . . . 21 Solar Constant

31 31 33 34 36 37 38 39 40 42 44 46 46 48 50 52

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22 Miscellaneous Physical Constants

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1 Kinematics Equations

Kinematics deals with problems involving distance, velocity, time and constant acceleration. The restriction that acceleration is a constant for these problems limits the scope of this subject, but a large body of applications remains. Vector concepts are not generally employed, so that velocity is undirected and equivalent to speed. Distance, denoted by x, refers to the total distance travelled, not necessarily the distance between the starting and stopping points. Force and mass are not involved in the kinematics relations. The ﬁrst equation relates the distance covered by an object during some time interval. Since the acceleration may be non-zero, the velocity may vary during the time interval under consideration. The most useful relation is x v= , (1.1) t where v is the average velocity, x is the total distance and t is the elapsed time. Given that acceleration is to be constant, velocity may be uniformly increasing or decreasing. A plot showing the case of increasing velocity is shown in Fig. (1.1). v v0 velocity (v − v0 )

time

t

Figure 1.1: Velocity Under Constant Acceleration The relation between acceleration and velocity is v − v0 , t v = v0 + at, a = 5...

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