Question

a particle moves along the x - axis so that at any time t ≥ 0 its position is given by x(t)=\frac{1}{2}(a - t)^{2}, where a is a positive constant. for what values of t is the particle moving to the right?
a the particle is moving to the right only if 0 < t < a.
b the particle is moving to the right only if a < t.
c the particle is moving to the right only if t = a.
d the particle is not moving to the right.

Explanation:

Step1: Find the velocity function

The velocity \(v(t)\) is the derivative of the position function \(x(t)\). Given \(x(t)=\frac{1}{2}(a - t)^{2}\), using the chain rule \((u^{n})^\prime=nu^{n - 1}u^\prime\) (here \(u=a - t\), \(n = 2\)), we have \(x^\prime(t)=(a - t)\times(- 1)=t - a\).

Step2: Determine when the particle is moving to the right

A particle is moving to the right when \(v(t)>0\). Set \(v(t)=t - a>0\). Solving the inequality \(t - a>0\) gives \(t>a\).

Answer:

B. The particle is moving to the right only if \(a < t\)