Kinematics of a Particle in a Plane (using polar description, example continued)

Recall v_P = 3i + 6 j ft/sec and a_P = 18 j ft/sec²

Strategy: Use the dot product of these vectors with unit vectors in the radial and transverse

directions to find the radial and transverse components of velocity and acceleration of the particle. Start with the position vector, r, of particle P at (1,1). r = i + j . | r | = √2 and e_r = r / | r |

So the unit vectors are:

e_r = (1/√2) ( i + j) and since e_θ is normal to e_r and in the direction of increasing θ

e_θ = (1/√2) (˗ i + j)

Now the component of velocity in the radial direction is

v_r = v_P ● e_r = (3i + 6 j) ● (1/√2) ( i + j)

So v_r = 9/√2 ft/sec and component of velocity in the transverse direction is

v_{θ =} v_P ● e_θ = (3i + 6 j) ● (1/√2) ( - i + j) = 3/√2 ft/sec

Likewise a_{r =} a_P ● e_r = 18j ● ((1/√2) ( i + j) = 18/√2 ft/sec² and

a_{θ =} a_P ● e_θ = 18j ● ((1/√2) ( - i + j) = 18/√2 ft/sec²

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