Q4. A motorist on an open ground turns 60° to the left after every 500 m (NCERT Exercise 3.10, page 47). Starting from a turn, find the displacement and the path length at the third, sixth and eighth turns.
His path traces a regular hexagon of side 500 m. At the third turn he is at the opposite corner: displacement 1000 m at 60° to the left of his starting direction, path 1500 m. At the sixth turn he is back at the start: displacement 0, path 3000 m. At the eighth turn he is where he was at the second: displacement 500√3 ≈ 866 m at 30° to the left of his starting direction, path 4000 m.
Q5. True or false, with reasons (NCERT Exercise 3.21, page 48): a scalar quantity (a) is conserved in a process; (b) can never be negative; (c) must be dimensionless; (d) does not vary from point to point; (e) has the same value for observers with differently oriented axes.
(a) False: kinetic energy, a scalar, is not conserved in an inelastic collision. (b) False: a temperature in °C or a potential energy can be negative. (c) False: mass is a scalar with dimension [M]. (d) False: temperature varies from place to place in a room. (e) True: a scalar has no direction, so turning the axes cannot change it.
Q6. If the origin is shifted by a fixed vector without rotating the axes, do position vectors change? Does the displacement between two points?
Every position vector changes, because each is drawn from the origin. The displacement does not: it is the arrow from P to P′, which joins the same two points wherever the origin is placed.