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wirewolf October 8th, 2016 02:20 PM

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?

skeeter October 8th, 2016 03:56 PM

$u = \dfrac{2t}{(1+t^2)^2}$

sub $t = \tan\left(\dfrac{x}{2}\right)$ ...

$u = \dfrac{2\tan\left(\dfrac{x}{2}\right)}{\left[1+\tan^2\left(\dfrac{x}{2}\right)\right]^2}$

$u = \dfrac{2\tan\left(\dfrac{x}{2}\right)}{\sec^4\left (\dfrac{x}{2}\right)}$

$u = 2\tan\left(\dfrac{x}{2}\right) \cos^4\left(\dfrac{x}{2}\right)$

$u = \color{red}{2\sin\left(\dfrac{x}{2}\right) \cos\left(\dfrac{x}{2}\right)} \color{blue}{ \cos^2\left(\dfrac{x}{2}\right)}$

$u = \color{red}{\sin{x}} \cdot \color{blue}{\dfrac{1+\cos{x}}{2}}$

skipjack October 9th, 2016 10:10 AM

To find f in terms of x, one can use f = du/dt = (du/dx)/(dt/dx).

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