Answered: move first with the acceleration…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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$**Motion of Particles A and B Along a Straight Line** Particles A and B start from rest at the origin $ s = 0 $ and move along a straight line. Particle A begins to move first with the acceleration profile given by: \[ a_A = (7t_A - 10 \cdot \cos(t)) \; \text{m/s}^2 \] Here, $ t_A $ is the time in seconds after particle A begins to accelerate. After one second had passed since particle A began to accelerate, particle B starts to move. The acceleration of particle B is described by the function: \[ a_B = (t_B^2 - 8) \; \text{m/s}^2 \] where $ t_B $ is the time in seconds after particle B has started to accelerate. The task is to find the absolute distance between particles A and B when $ t_A = 5 $ seconds. **Procedure:** 1. **Acceleration of Particle A:** \[ a_A = 7t_A - 10 \cdot \cos(t) \] 2. **Acceleration of Particle B:** \[ a_B = t_B^2 - 8 \] 3. **Determine velocities by integrating the acceleration functions:** For Particle A: \[ v_A = \int a_A \, dt = \int (7t_A - 10 \cdot \cos(t)) \, dt \] For Particle B: \[ v_B = \int a_B \, dt = \int (t_B^2 - 8) \, dt \] 4. **Determine positions by integrating the velocity functions:** For Particle A: \[ s_A = \int v_A \, dt \] For Particle B: \[ s_B = \int v_B \, dt \] 5. **Find the position of each particle at $ t_A = 5 $ seconds and determine the absolute distance between them.** This involves solving the integrals to obtain the velocity and position functions, then evaluating these functions at the given times to find the positions of the particles. Subtract the positions to determine the distance between them. **Note:** It is crucial to appropriately adjust $ t_B $ since particle B starts accelerating 1 second after particle A. Thus, when $ t_A = 5 $ seconds, \( t$

Transcribed Image Text:**Motion of Particles A and B Along a Straight Line** Particles A and B start from rest at the origin $ s = 0 $ and move along a straight line. Particle A begins to move first with the acceleration profile given by: \[ a_A = (7t_A - 10 \cdot \cos(t)) \; \text{m/s}^2 \] Here, $ t_A $ is the time in seconds after particle A begins to accelerate. After one second had passed since particle A began to accelerate, particle B starts to move. The acceleration of particle B is described by the function: \[ a_B = (t_B^2 - 8) \; \text{m/s}^2 \] where $ t_B $ is the time in seconds after particle B has started to accelerate. The task is to find the absolute distance between particles A and B when $ t_A = 5 $ seconds. **Procedure:** 1. **Acceleration of Particle A:** \[ a_A = 7t_A - 10 \cdot \cos(t) \] 2. **Acceleration of Particle B:** \[ a_B = t_B^2 - 8 \] 3. **Determine velocities by integrating the acceleration functions:** For Particle A: \[ v_A = \int a_A \, dt = \int (7t_A - 10 \cdot \cos(t)) \, dt \] For Particle B: \[ v_B = \int a_B \, dt = \int (t_B^2 - 8) \, dt \] 4. **Determine positions by integrating the velocity functions:** For Particle A: \[ s_A = \int v_A \, dt \] For Particle B: \[ s_B = \int v_B \, dt \] 5. **Find the position of each particle at $ t_A = 5 $ seconds and determine the absolute distance between them.** This involves solving the integrals to obtain the velocity and position functions, then evaluating these functions at the given times to find the positions of the particles. Subtract the positions to determine the distance between them. **Note:** It is crucial to appropriately adjust $ t_B $ since particle B starts accelerating 1 second after particle A. Thus, when $ t_A = 5 $ seconds, \( t

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