The path rlt-It-sin t) i+ (1-cos t) j desc nbes motion on the cycloid x : t-sin

Question

The path rlt-It-sin t) i+ (1-cos t) j desc nbes motion on the cycloid x : t-sin t, y: 1-cos t. Find the particle's velocity and acceleration vectors at t : ketch them as vectors on the curve and i+ The velocity vector a Simplity your answer, including any radicals. Use integers or fractions for any numbers in the expression) The acceleration vector at "Z is. . i+ D Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression) Choose the correct sketch for the particle's velocity and acceleration att- OA, O B O C

Explanation / Answer

From the given question,

x=t-sint, y=1-cost

r= xi+ yj

velocity v= dr/dt = dx/dt i + dy/dt j

=(1-cost)i + (sint)j

v( pi/2) = (1- cos (pi/2) )i + ( sin(pi/2)) j

v(pi/2)=i +j

acceleration a = dv/dt = sint i + cost j

a(pi/2) = sin(pi/2) i + cos(pi/2) j

a(pi/2)=i

Among the sketches given, option (a) is correct. as direction of velocity is tan^-1 (1/1) = 45 degrees.

direction of acceleration is along x axis

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