A clock has an hour hand of length 2.4 cm and a minute hand of length 3.8 cm. (a) Calculate the position and velocity of the hour hand at noon. (b) Calculate the position and velocity of the minute hand at 12:15.
At 12:00 the hour hand it's located at 0 degrees and its tip travels with a velocity of 1.26 cm/hour, approximately.
At 12:15 the minute hand it's located at 90 degrees and its tip travels with a velocity of 23.88 cm/hour, approximately.
Explanation:
Position is given by an angle, in this case starting from the 12:00 mark and opening clockwise.
At noon, the the hour hand is located at the 12:00 mark, so its position is 0 degrees; and it has an angular velocity of (2π)/(12 hour) = π/(6 hour). So its tip travels with linear velocity of (2.4 cm) * π/(6 hour) ≅ 1.26 cm/hour.
At 12:15, the minute hand is located at the 3 hour mark, so its position is 90 degrees or π/2; and it has an angular velocity of (2π)/(1 hour) = 2π/hour. So its tip travels with linear velocity of (3.8 cm) * 2π/hour ≅ 23.88 cm/hour.
