### Particle Motion in the XY Plane with Constant Acceleration**Problem Statement:**1. A particle starts from the origin at \( t = 0 \) with a velocity of \((16\mathbf{i} - 12\mathbf{j}) \, \text{m/s}\) and moves in the \( xy \)-plane with a constant acceleration of \(\mathbf{a} = (3.0\mathbf{i} - 6.0\mathbf{j}) \, \text{m/s}^2\). a. What is the position vector at any time \( t \)? b. Determine the direction (as measured from the positive \( x \)-axis, counterclockwise) of the position at \( t = 2s \).**Solution:**a. **Position Vector at Any Time \( t \):** The position vector \(\mathbf{r}(t)\) of a particle moving with initial velocity \(\mathbf{v_0}\) and constant acceleration \(\mathbf{a}\) can be found using the kinematic equation: \[ \mathbf{r}(t) = \mathbf{r_0} + \mathbf{v_0} t + \frac{1}{2} \mathbf{a} t^2 \] Given: - Initial position \(\mathbf{r_0} = 0\) - Initial velocity \(\mathbf{v_0} = 16\mathbf{i} - 12\mathbf{j} \, \text{m/s}\) - Acceleration \(\mathbf{a} = 3.0\mathbf{i} - 6.0\mathbf{j} \, \text{m/s}^2\) Substituting these values in the equation: \[ \mathbf{r}(t) = (16\mathbf{i} - 12\mathbf{j})t + \frac{1}{2}(3.0\mathbf{i} - 6.0\mathbf{j})t^2 \] Simplifying: \[ \mathbf{r}(t) = (16t + \frac{3}{2}t^2) \mathbf{i} + (-12t - 3t^2) \mathbf{j} \]

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Description: ### Particle Motion in the XY Plane with Constant Acceleration**Problem Statement:**1. A particle starts from the origin at \( t = 0 \) with a velocity of \((16\mathbf{i} - 12\mathbf{j}) \, \text{m/s}\) and moves in the \( xy \)-plane with a constan

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4. A particle starts from the origin at t=0 with a velocity of (161 - 12j) m/s and moves in the xy plane with a constant acceleration of a= (3.01 – 6.0ĵ) m/s². a. What is the position vector at any time t? b. Determine the direction (as measured from the positive x axis, counterclockwise) of the position at t=2s.

4. A particle starts from the origin at t=0 with a velocity of (161 - 12j) m/s and moves in the xy plane with a constant acceleration of a= (3.01 – 6.0ĵ) m/s². a. What is the position vector at any time t? b. Determine the direction (as measured from the positive x axis, counterclockwise) of the position at t=2s.

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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