By Kleppner D., Kolenkow R.
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2 Without loss of generality, we can let r0 = 0, and assume that v0y = 0. ) Then x = v0x t 1 z = v0z t − gt2 . e. the path z(x). v0z g z= x − 2 x2 . v0x 2v0x z v0 x As shown in the sketch, this is the well-known parabola of projectile motion under constant gravity. 11 The Effect of a Radio Wave on an Ionospheric Electron The ionosphere is a region of electrically neutral gas, composed of positively charged ions and negatively charged electrons, that surrounds the Earth at a height of approximately 200 km (120 mi).
We therefore write dx , v= dt using notation due to Gottfried Leibniz, who independently invented calculus. Newton would have written v = x˙ where the dot stands for d/dt. Following a convention frequently used in physics, we shall use Newton’s notation only for derivatives with respect to time. The derivative of a function f (x) can also be written f (x) ≡ d f (x)/dx. In a similar fashion, the instantaneous acceleration a is v(t + Δt) − v(t) a = lim Δt→0 Δt dv = v˙ . 8 VELOCITY AND ACCELERATION 15 Using v = dx/dt, a= d2 x = x¨.
U cos θ θ x As the particle moves to the right, θ decreases and rˆ and θˆ change direction. Ordinarily, of course, we try to use coordinates that make the problem as simple as possible; polar coordinates can be used here, but they are not well suited to this problem. 16 Velocity of a Bead on a Spoke A bead moves along the spoke of a wheel at constant speed u meters per second. The wheel rotates with uniform angular velocity θ˙ = ω radians per second about an axis fixed in space. ω u At t = 0 the spoke is along the x axis, and the bead is at the origin.
An Introduction to Mechanics by Kleppner D., Kolenkow R.