The particle will come to rest when its velocity is zero. To find the velocity, we differentiate the position function with respect to time:
v=‌=a√λ‌cos(√λt+b)Setting the velocity to zero, we get:
a√λ‌cos(√λt+b)=0This equation is satisfied when:
cos(√λt+b)=0The cosine function is zero at odd multiples of
‌. Therefore, we have:
√λt+b=(2n+1)‌, where
n is an integer.
Solving for
t, we get:
t=‌ This gives us an infinite number of times when the particle comes to rest. Let's consider the first two instances (
n=0 and
n=1 ):
‌t1=‌‌t2=‌ Now, we can find the positions of the particle at these times by plugging them back into the original position function:
‌x1=asin‌(√λt1+b)=asin‌(‌)=a‌x2=asin‌(√λt2+b)=asin‌(‌)=−aThe distance between these two points is:
|x2−x1|=|−a−a|=2aTherefore, the correct answer is Option C: 2a.