Calculus word problem question

Question

A particle moves along the x-axis with an acceleration according to the formula a(t)=cos(t)-sin(2t), where t is measured in seconds and displacement is measured in meters. If at time t=0 the position of the particle is 3 meters (i.e., v(0)=1.5), find the position of this particle at t=( π/2).

Answer 1

To solve this problem, we can integrate a(t) twice with respect to time to obtain position x(t).

 

v(t) = ∫a(t)dt = ∫[cos(t) - sin(2t)]dt = sin(t) + (1/2)cos(2t) + c₁

 

Given v(0) = 1.5,

 

1.5 = sin(0) + (1/2)cos(0) + c₁ = 0 + 0.5 + c₁ ⇒ 1.5 = 0.5 + c₁ ⇒ c₁ = 1.5 - 0.5 = 1

 

v(t) = sin(t) + (1/2)cos(2t) + 1

 

x(t) = ∫v(t)dt = ∫[sin(t) + (1/2)cos(2t) + 1]dt = -cos(t) + (1/4)sin(2t) + t + c₂

 

Given x(0) = 3,

 

3 = -cos(0) + (1/4)sin(0) + 0 + c₂ ⇒ 3 = -1 + c₂ ⇒ c₂ = 4

 

x(t) = -cos(t) + (1/4)sin(2t) + t + 4

 

x(π/2) = -cos(π/2) + (1/4)sin(2π/2) + π/2 + 4 = 0 + 0 + π/2 + 4

 

x(π/2) = (4 + π/2) meters