• Jan 9th 2009, 09:10 AM
mattty
The distance, d (in metres), which an object travels. is given by the equation \$\displaystyle d=ut-8t^2 \$ where u is the initial velocity (in metres per second) and t is the time in seconds it has been travelling. Show that there is only one value for t when \$\displaystyle u^2 =32d\$

I'm having real trouble solving this equation for t using any algebraic method that I've learnt for quadratic equations.
I can't figure out how to put it into the quadratic formula simply for the fact that i can't get any of the variables by themself on either side of the equation to set as x.

The steps I've tried so far in order to rearrange for t are:
\$\displaystyle d=ut-8t^2\$
\$\displaystyle d+8t^2=ut\$
divide both sides by t
\$\displaystyle d/t+8t=u\$
take d/t away from both sides
\$\displaystyle 8t=u-d/t\$
and here is where i lose track, because i figure out that i can't rearrange for t at all. i've considered converting d/t into V (final velocity) but i can't see that helping much.
Any suggestions would be much appreciated.
• Jan 9th 2009, 09:59 AM
Constatine11
Quote:

Originally Posted by mattty
The distance, d (in metres), which an object travels. is given by the equation \$\displaystyle d=ut-8t^2 \$ where u is the initial velocity (in metres per second) and t is the time in seconds it has been travelling. Show that there is only one value for t when \$\displaystyle u^2 =32d\$

.

Rearrange the expression for \$\displaystyle d\$:

\$\displaystyle
8t^2-ut+d=0
\$

This is an equation relating \$\displaystyle t,\ u\$ and \$\displaystyle d\$ .

If for some particular \$\displaystyle d\$ and \$\displaystyle u\$ we wish to solve for \$\displaystyle t\$ we will use the quadratic formula. Now the discriminant for the quadratic is:

\$\displaystyle
D=u^2-32d
\$

and it is well known that we have no real roots for \$\displaystyle t\$ if \$\displaystyle D<0\$, two real roots if \$\displaystyle D>0\$, and exactly one real root if \$\displaystyle D=0\$.

The last condition above is the one applicable here.

.
• Jan 9th 2009, 07:35 PM
mattty
Many thanks for this, apparently maths at 3am isn't my strong point :D