Physics

Physics · mechanics

Problem

iii. What is the minimal launching speed

v_{\min }

needed to hit the topmost point of a spherical building of radius

R

? <image_3> La Geode, Parc de la Villette, Paris. Photo: katchooo/flickr.com Given constants: Problem T1. Focus on sketches Part A. Ballistics A ball, thrown with an initial speed

v_{0}

, moves in a homogeneous gravitational field in the

x-z

plane, where the

x

-axis is horizontal, and the

z

-axis is vertical and antiparallel to the free fall acceleration

g

. Neglect the effect of air drag. Context question: i. By adjusting the launching angle for a ball thrown with a fixed initial speed

v_{0}

from the origin, targets can be hit within the region given by

z \leq z_{0}-k x^{2}

You can use this fact without proving it. Find the constants

z_{0}

and

k

. Context answer:

\frac{v_{0}^{2}}{ 2 g}

\frac{g}{2 v_{0}^{2}}

Context question: ii. The launching point can now be freely selected on the ground level

z=0

, and the launching angle can be adjusted as needed. The aim is to hit the topmost point of a spherical building of radius

R

(see fig.) with the <image_1> minimal initial speed

v_{0}

. Bouncing off the roof prior to hitting the target is not allowed. Sketch qualitatively the shape of the optimal trajectory of the ball (use the designated box on the answer sheet). Note that the marks are given only for the sketch. Context answer: <image_2>

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