<p>A particle moves along the x axis. Its position is given by the equation x = 1.8 + 2.5t − 3.6t^ 2 with x in meters and t in seconds.</p>
<p>(a) Determine its position when it changes direction.
(b) Determine its velocity when it returns to the position it had at t = 0? (Indicate the direction of the velocity with the sign of your answer.)
plz explain with details and thanks</p>
<p>For the first part, I found the derivative of the position function given because this is the function for velocity. It was v(t) = 2.5 - 7.2t. Then I set that equal to zero because velocity equals zero when the particle changes direction. I found that t = .35 when velocity is zero.</p>
<p>The second part was harder, but I think I have the right answer now. First, you need to find what the position was at t = 0. Just plug 0 into the position function and you will get 1.8. Then you need to find at what other time x = 1.8, other than 0. I set the position function equal to 1.8 to solve for t. This gave me 1.8 = 1.8 + 2.5t - 3.6t^2. Then, I subtracted 1.8 from both sides to get 0 = 2.4t - 3.6t^2. Next, I factored out a t to get 0 = t(2.5 - 3.6t). I solved this to find t = 0 and t = .694. I knew I wanted to find velocity at .694, so I plugged this into the velocity function that I already found in the first part and got -2.4968.</p>
<p>An alternate approach to this problem, one that does not require calculus, is to observe that the equation for the particle’s position is that for an object moving at constant acceleration.</p>
<p>The general equation, perhaps familiar to you from the kinematics of an object in a gravitational field, is:</p>
<p>x(t) = x(0) + v(0)t + (1/2) at^2</p>
<p>So x(0) = 1.8, v(0) = 2.5, and a = -3.5</p>
<p>In the x-t coordinate system this is the equation for a parabola.</p>
<p>The position at which the particle changes position is when x(t) is at its maximum. Recall that for a parabola that’s at (-2.5/(2*(-3.5)). [-B/2A]</p>
<p>When the particle returns to its initial position it has the same speed that it did when it started off except it is now moving in the opposite direction – so v= -v(0) = -2.5. This is intuitive because the parabola is symmetric about the maximum value of x.</p>
<p>There’s a typo in my solution.</p>
<p>The time at which the particle changes position is at (-2.5/(2*(-3.5)). I stated that this is the position. That’s not correct. To get the position substitute the time (above) in the equation for x.</p>
<p>Is this a homework problem?</p>