Question

Supposing d(t) is known to have value D,
what procedure will find the time t at which
this happens?

Answer 1

Answer:

• The procedure is: solve the quadratic equation for $t$.

Explanation:

This question assumes uniformly accelerated motion, for which the distance d a particle travels in time t is given by the general equation:

• $d(t)=d_0+v_0t+at^2/2$

That is a quadratic equation, where the independent variable is the time $t$.

Thus, the procedure that will find the time t at which the distance value is known to be D is to solve the quadratic equation for $t$.

To solve it you start by changing the equation to the general form of the quadratic equations, rearranging the terms:

• $(a/2)t^2+v_0t+(d_0-D)=0$

Some times that equation may be solved by factoring, and always it can be solved by using the quadratic formula:

• $t=\frac{-b+/-\sqrt{b^2-4ac} }{2a}$

Where:

$a=-a/2\\ \\ b=v_0\\ \\ c=d_0-D$

That may have two solutions. Some times one of the solution makes no physical sense (for example time cannot be negative) but others the two solutions are valid.