Physics Technology Update (4th Edition)

Published by Pearson
ISBN 10: 0-32190-308-0
ISBN 13: 978-0-32190-308-2

Chapter 4 - Two-Dimensional Kinematics - Problems and Conceptual Exercises - Page 106: 35

Answer

(a) They are the same (b) In both cases we get $v_f=17.5\text{ m/s}$

Work Step by Step

(a) Snowballs will land with the same speed. This is due to the conservation of energy law. Namely, their initial energy (the sum of the potential and kinetic energy) is the same, so their final energy (which is only kinetic and depends on the speed only) will also be the same. (b) The first ball accelerates downwards with the acceleration of gravity and no initial vertical component of the velocity so the final vertical velocity component will be $$v_{yf}=\sqrt{2gh}.$$ The horizontal component remains constant (and thus equal to the initial velocity) so we have for the landing speed $$v_f=\sqrt{v_x^2+v_{yf}^2}=\sqrt{13^2+2\cdot9.81\cdot7.0}\text{ m/s}=17.5\text{ m/s}.$$ For the second ball, we have that it will reach the maximum height of $$h_{max}=h+\frac{v_{0y}^2}{2g}=7.0\text{ m}+\frac{(13\sin25^\circ)^2}{2\cdot9.81}\text{ m}=8.54\text{ m}.$$ Now, in that moment it will have the horizontal component of the velocity $v_{x}=v_0\cos25^\circ$ which will remain constant and no initial vertical component. Thus, the final vertical component is $$v_{yf}=\sqrt{2gh_{max}}$$ so the landing speed is $$v_f=\sqrt{v_x^2+v_{yf}^2}=\sqrt{(13\cos25^\circ)^2+2\cdot9.81\cdot8.54}\text{ m/s}=17.5\text{ m/s}.$$ In both cases we got the same value.
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