The kinematic equation for the coordinate x(t) of a moving particle

Question

The kinematic equation for the coordinate x(t) of a moving particle is given in the following implicit form: x2 – (t+1)2 = 3 The initial values are: x(0) = 2 and vx(0) = 0.5 as t = 0. Find the instant velocity vx(t) and for the moment t = 2.

Answer 1

I'm unsure if you have been taught how to take derivatives, but I will assume you have. If not, please let me know and I'll answer the question differently.

Velocity is the derivative of position so V = dx/dt

Since x² - (t + 1)² = 3 is not a typical function written as x(t) = (some function having "t" as the variable) then we can take the derivative implicitly to find velocity. That means you are taking the derivative of each part with respect to time.

First, take the derivative of x² with respect to time. Notice that x² does not have "t" in it as written although we know that the position "x" does change with respect to time in some way, we just do not know exactly how it changes (square root or ??) so we apply the chain rule and we take the derivative with respect to "x" (which is "2x") then multiply by the derivative of "x" with respect to time (which is dx/dt). So the derivative of x² with respect to time is (2x)(dx/dt).

Then we take the derivative of - (t + 1)2 with respect to "t" which is (by the power rule): -2(t + 1).

Next take the derivative of "3" with respect to time which is 0 (the derivative of a constant is zero).

Putting these together:

(2x)(dx/dt) - 2(t + 1) = 0

(2x)(dx/dt) = 2(t + 1)

dx/dt = 2(t + 1)/2x

V = dx/dt = (t + 1)/x

Notice that this equation requires you to have both "t" and "x" to find the velocity. So we must find "x". We can use the position equation and plug in t = 2 and solve for "x":

x² - (t + 1)² = 3

x² - ((2) + 1)² = 3

x² - (3)² = 3

x² - 9 = 3

x² = 12

x = ±√12 = ±3.464

So, we have 2 possible values of "x". I suppose you could look at the initial value of x where at t = 0, x = 2 and since x² - ((0) + 1)² = 3 solves to x² = 4 and since you were given x = 2 as the result that we could ASSUME the positive √12 is the velocity. But I think that is very presumptuous. I'll leave it to you if you'd like to do that.

So to find V, plug in t = 2 and x = either +√12 or -√12 or both to get the final answer:

V = ((2) + 1)/√12 = 0.866

V = ((2) + 1)/(-√12) = -0.866