A ball is thrown straight up from the ground with initial speed

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles

Mathematical Methods for Physics, Ch Delta functions, distributions and generalised functions

Problem

A ball is thrown straight up from the ground with initial speed v0=9.8 m s−1v_0 = 9.8\,\mathrm{m\,s^{-1}}, so its height is y(t)=v0t−12gt2y(t) = v_0t - \tfrac12 g t^2 with g=9.81 m s−2g = 9.81\,\mathrm{m\,s^{-2}}. A photogate at ground level outputs a voltage V(t)=V0 δ(y(t))V(t) = V_0\,\delta(y(t)), with V0=5.0 V mV_0 = 5.0\,\mathrm{V\,m}, whenever the ball is at y=0y=0. ((a)) Using δ(y(t))=∑iδ(t−ti)/∣y′(ti)∣\delta(y(t)) = \sum_i \delta(t-t_i)/|y'(t_i)| over the simple zeros of yy, and considering only the landing instant t1>0t_1>0 (ignore the launch instant t=0t=0), find t1t_1 and write V(t)V(t) near t1t_1 as a coefficient times δ(t−t1)\delta(t-t_1). ((b)) State that coefficient, with its unit.

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