Calculus Calculus Math Forum

 October 8th, 2016, 02:20 PM #1 Member   Joined: Mar 2015 From: uk Posts: 33 Thanks: 1 distance speed and acceleration I'm struggling with this one; can anyone help? A body moves in a straight line with the equation s = t^2 / (1+(t^2)) where t is time in seconds and s is distance travelled in metres. Show that s is greater than or equal to 0 and less than 1. Show also that speed u and acceleration f are given at time t by u = sinx (1+cosx)/2 f = (2cosx - 1) (1 + cosx)^2 / 2 where t = tan(x/2) I think I've got the first part by using limits. As t tends to zero then we have 0 / (1+0) = 0 and as t tends to infinity we have (infinity / (1+infinty) which tends to 1. For the second part, speed is rate of change of distance with time or ds/dt. Differentiating using the quotient rule gives ds/dt = 2t / ((1+(t^2))^2) I've then substituted tan(x/2) for t and tried using trig identities to get the answer but without success. Can anyone help with where I'm going wrong? Last edited by skipjack; October 9th, 2016 at 09:48 AM. October 8th, 2016, 03:56 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,947 Thanks: 1555 $u = \dfrac{2t}{(1+t^2)^2}$ sub $t = \tan\left(\dfrac{x}{2}\right)$ ... $u = \dfrac{2\tan\left(\dfrac{x}{2}\right)}{\left[1+\tan^2\left(\dfrac{x}{2}\right)\right]^2}$ $u = \dfrac{2\tan\left(\dfrac{x}{2}\right)}{\sec^4\left (\dfrac{x}{2}\right)}$ $u = 2\tan\left(\dfrac{x}{2}\right) \cos^4\left(\dfrac{x}{2}\right)$ $u = \color{red}{2\sin\left(\dfrac{x}{2}\right) \cos\left(\dfrac{x}{2}\right)} \color{blue}{ \cos^2\left(\dfrac{x}{2}\right)}$ $u = \color{red}{\sin{x}} \cdot \color{blue}{\dfrac{1+\cos{x}}{2}}$ Thanks from topsquark and wirewolf October 9th, 2016, 10:10 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,747 Thanks: 2133 To find f in terms of x, one can use f = du/dt = (du/dx)/(dt/dx). Thanks from topsquark Tags acceleration, distance, speed Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post sanjanagaur Elementary Math 9 September 11th, 2016 11:27 AM szz Physics 2 December 17th, 2014 10:12 PM shanti Physics 2 September 21st, 2014 08:13 PM MathsIlove Elementary Math 8 September 4th, 2014 07:28 PM krieve Algebra 1 September 26th, 2008 04:51 AM

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