Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 11 - Work - Exercises and Problems: 15

Answer

At x = 1 m, the velocity is 3.7 m/s At x = 2 m, the velocity is 6.6 m/s At x = 3 m, the velocity is 9.7 m/s

Work Step by Step

The work done by the force is equal to the area under the force versus position graph. We can find the work done by the force from x = 0 to x = 1 m. $W = \frac{1}{2}(5~N)(1~m)$ $W = 2.5~J$ We can use the work-energy theorem to find the velocity at x = 1 m; $KE_f = KE_0+W$ $\frac{1}{2}mv_f^2 = \frac{1}{2}mv_0^2+W$ $v_f^2 = \frac{mv_0^2+2W}{m}$ $v_f = \sqrt{\frac{mv_0^2+2W}{m}}$ $v_f = \sqrt{\frac{(0.50~kg)(2.0~m/s)^2+(2)(2.5~J)}{0.50~kg}}$ $v_f = 3.7~m/s$ At x = 1 m, the velocity is 3.7 m/s. We can find the work done by the force from x = 0 to x = 2 m. $W = \frac{1}{2}(10~N)(2~m)$ $W = 10~J$ We can use the work-energy theorem to find the velocity at x = 2 m; $KE_f = KE_0+W$ $\frac{1}{2}mv_f^2 = \frac{1}{2}mv_0^2+W$ $v_f^2 = \frac{mv_0^2+2W}{m}$ $v_f = \sqrt{\frac{mv_0^2+2W}{m}}$ $v_f = \sqrt{\frac{(0.50~kg)(2.0~m/s)^2+(2)(10~J)}{0.50~kg}}$ $v_f = 6.6~m/s$ At x = 2 m, the velocity is 6.6 m/s. We can find the work done by the force from x = 0 to x = 3 m. $W = \frac{1}{2}(15~N)(3~m)$ $W = 22.5~J$ We can use the work-energy theorem to find the velocity at x = 3 m; $KE_f = KE_0+W$ $\frac{1}{2}mv_f^2 = \frac{1}{2}mv_0^2+W$ $v_f^2 = \frac{mv_0^2+2W}{m}$ $v_f = \sqrt{\frac{mv_0^2+2W}{m}}$ $v_f = \sqrt{\frac{(0.50~kg)(2.0~m/s)^2+(2)(22.5~J)}{0.50~kg}}$ $v_f = 9.7~m/s$ At x = 3 m, the velocity is 9.7 m/s.
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