I'm a student preparing for my mathematics finals and I have come across a calculus related question which has proven to be quite confusing for me So could you kindly explain to me in detail the solution to this problem
Q) A particle moves along a straight line such that it's displacement , X metres, from a fixed point O of the line at the time t seconds is given by X=t-16/t , for t is equal or greater than 1
A) find the value of t when the particle is at O
B) The velocity in m/s of the particle when t=5
We have been given an equation which gives us the relationship between the position of the particle X, as measured from the fixed point O, and the time t.
X = t - 16/t
The problem states that this equation is only valid for values of the time t that are greater than or equal to 1. We may not need to worry about that information in this problem but it is nevertheless good practice to notice it.
A) Now, the first question we need ask ourselves is the value of X when the particle is at O. This is obviously X = 0 (zero). We insert this into the equation and find the corresponding value of t.
0 = t - 16/t
t = 16/t
multiplying both sides by t;
t² = 16
t = ±4
There are two values of t that satisfy the equation, but notice that one of them is NOT greater than or equal to 1, as the equation requires in order to be valid, and so we have to discard that value. The correct and only answer is then t = 4 seconds. It turns out we needed to use that extra information about the equation.
B) We know from calculus that the velocity at time t is given by
V = dX/dt
= 1 - (-16/t²)
= 1 + 16/t²
And at t = 5, we have
V = 1 + 16/5²
= 1 + 16/25
= 1 + 0.64
The velocity at t = 5 seconds is therefore 1.64 meter/seconds.