Question

A particle moves along the x-axis according to the law x = acosωt. Find the distance that the particle covers during the time interval from t = 0 to t.

Answer 1

From the motion law, x = acosωt,it is obvious that the time taken to cover the distance equal to the amplitude (a), starting from extreme position equal T/4.

Now one can write 

As the particle moves according to the law, x = acosωt,

so at n = 1,3,5.... or for odd n values it passes through the mean position and for even numbers of n it comes to an extreme position(if t₀ = o).

case (1) when n is an odd number :

in this case, from the equation 

x = ± asinωt, if the t is counted from nT/4 and the distance covered in the time interval to becomes,s₁ = asinωt₀ = a sinω(t - n(T/4)) = asin(ωt - nπ/2)

Thus the sought distance covered for odd n is 

Case (2), When n is even, in this case from the equation 

x = acosωt,the distance covered (s₂) in the interval t₀, is given by 

Hence the sought distance for n is even