Problem 3: A mass m is thrown from the origin att = 0 with initial three-momentum po in the y direction. If it is subject to a constant force Fo in the x direction, find its velocity v as a function of t, and by integrating v find its trajectory. Check that in the non-relativistic limit the trajectory is the expected parabola. Hint: The relationship F = P is still true in relativistic mechanics, but now p = ymv instead of p = mv. To find the non-relativistic limit, treat c as a very large quantity and use the Taylor approximation (1+ x)" = 1 + nx when a is small.

Consider the equation for kinetic energy: KE = 1/2mv^2 = 1/2 * m * v^2. If I ask you to take the derivative of kinetic energy, you should ask "the derivative with respect to what?" a) Suppose mass m is constant. Compute the derivative of KE with respect to v, (d(KE)/dv).   b) Who takes derivatives with respect to velocity? No one. Except you, just now. Sorry. The rate of change of energy with respect to time is more important: it is the Power. Now, consider velocity v to be a function of time, v(t). We will rewrite KE showing this time dependance: KE= 1/2 * m * v(t)^2. Show that (d(KE)/dt) = F(t)v(t). Hint: use Newton's second law, F = ma, to simplify. c) In the computation above, we assumed m was constant, and v was changing in time. Think of a physical situation in which both m and v are varying in time.  d) Compute the Power when both mass and velocity are changing in time. (First rewrite KE(t) showing time dependence, then compute (d(KE)/dt).

A particle of mass M, which was initially at rest, is hit by a particle of mass m, which is travelling at the speed Vi. The two particles collide and travel in different directions at the same pace Vf. Find the mass ratio M/m in terms of ß = vi/c, assuming a relativistic, elastic collision. Demonstrate that this reduces to Vi 0, the nonrelativistic meaning.

Calculate the interval ∆s2 between two events with coordinates ( x1 = 50 m, y1 = 0, z1 = 0,t1 = 1 µs) and (x2 = 120 m, y2 = 0, z2 = 0, t2 = 1.2 µs) in an inertial frame S. b) Now transform the coordinates of the events into the S'frame, which is travelling at 0.6calong the x-axis in a positive direction with respect to the frame S. Hence verify that thespacetime interval is invariant.

(a) Find the momentum (in kg- m/s) of a 1.20 x 109 kg asteroid heading towards the Earth at 3,050 km/s. (Hint: Use the approximation that y = 1 + 2c² least six significant figures.) kg-m/s (b) Find the ratio of this momentum to the classical momentum. (Enter your answer to at least six significant figures.) Р Pclassical at low velocities. Enter your answer to at

Given the time series below. Which of these can be interpreted as a generic chaotic trajectory of some dynamical system? 1) Acos(b sin(t)) 2) A1cos(f1t)+A2sin(f2t) 3) Acos(at2 +b) 4) Atang[at/(1+t)]