[list=1][*]An object's initial position is [math]\vec{r_i} = 2\hat{i} - 5\hat{j}[/math] m. After 4 seconds, its position is [math]\vec{r_f} = 10\hat{i} + 3\hat{j}[/math] m. Find the average velocity vector during this time interval.[br][/*][*]A particle moves with a constant velocity of [math]\vec{v} = 3\hat{i} - 2\hat{j}[/math] m/s. If it starts at the origin, what is its position vector after 6 seconds?[br][/*][*]A ball is thrown vertically upward with an initial velocity of [math]v_{0y} = 15[/math] m/s. Assuming only gravity acts on it ( [math]\vec{a} = -10\hat{j}[/math] m/s[sup]2[/sup]), find the ball's velocity vector after 2 seconds.[br][/*][*]The acceleration of a rocket is given by [math]\vec{a}(t) = (0.5t)\hat{i} + (2)\hat{j}[/math] m/s[sup]2[/sup]. If the rocket starts from rest at the origin, find its velocity vector at t = 4 seconds.[br][/*][*]A car's velocity is given by [math]\vec{v}(t) = (5t - 2)\hat{i}[/math] m/s. Find the displacement of the car between t = 1 second and t = 3 seconds.[br][/*][*]An object has an initial velocity of [math]\vec{v_0} = -3\hat{i} + 1\hat{j}[/math] m/s and a constant acceleration of [math]\vec{a} = 2\hat{i} - 0.5\hat{j}[/math] m/s[sup]2[/sup]. Find its velocity vector after 4 seconds.[br][/*][*]A particle's acceleration is given by [math]\vec{a}(t) = (6t)\hat{i} - (2)\hat{j}[/math] m/s[sup]2[/sup]. If its initial velocity is [math]\vec{v_0} = 1\hat{i} + 3\hat{j}[/math] m/s, find its velocity vector as a function of time.[br][/*][*]The position vector of a moving object is [math]\vec{r}(t) = (t^3 - t)\hat{i} + (2t^2)\hat{j}[/math] m. Find its acceleration vector at t = 2 seconds.[br][/*][*]A projectile is launched with an initial velocity of [math]\vec{v_0} = 10\hat{i} + 20\hat{j}[/math] m/s from ground level. Ignoring air resistance, what is the y-component of its velocity at its maximum height? What is the acceleration vector at the maximum height?[br][/*][*]Identify and describe what is wrong with the following equation and provide a corrected version (if applicable, or explain why no correction is possible):[br][math]\vec{v} = \frac{d}{dt}x[/math], where [math]\vec{v}[/math] is velocity and [math]x[/math] is position.[br][/*][/list]

[list=1][*][math]2\hat{i} + 2\hat{j}[/math] m/s[br][/*][*][math]18\hat{i} - 12\hat{j}[/math] m[br][/*][*][math]-5\hat{j}[/math] m/s[br][/*][*][math]4\hat{i} + 8\hat{j}[/math] m/s[br][/*][*][math]16\hat{i}[/math] m[br][/*][*][math]5\hat{i} - 1\hat{j}[/math] m/s[br][/*][*][math](3t^2 + 1)\hat{i} + (-2t + 3)\hat{j}[/math] m/s[br][/*][*][math]12\hat{i} + 4\hat{j}[/math] m/s[sup]2[/sup][br][/*][*]0 m/s, [math]-10\hat{j}[/math] m/s[sup]2[/sup][br][/*][*]Position is a vector, not a scalar. It should be [math]\vec{v} = \frac{d}{dt}\vec{r}[/math].[br][/*][/list]