Motion in a Plane

4.11 A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi (b) the magnitude of average velocity? Are the two equal?

4.10 On an open ground, a motorist follows a track that turns to his left by an angle of 60 $°$ after every 500 m. Starting from a given turn; specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.

4.9 A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the

(a) Net displacement

(b) Average velocity

(c) Average speed of the cyclist?

4.8 Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of path skate?

4.7 Given a + b + c + d = 0, which of the following statements are correct:

(a) a, b, c, and d must each be a null vector

(b) The magnitude of (a + c) equals the magnitude of (b + d)

(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d

(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear?

4.6 Establish the following vector inequalities geometrically or otherwise:

(a) |a+b| < |a| + |b|

(b) |a+b| > a| −|b

(c) |a−−b| < |a| + |b|

(d) |a−−b| > a| − |b

When does the equality sign above apply?

4.5 Read each statement below carefully and state with reasons, if it is true or false:

(a) The magnitude of a vector is always a scalar

(b) Each component of a vector is always a scalar

(c) The total path length is always equal to the magnitude of the displacement vector of a particle

(d) The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time

(e) Three vectors not lying in a plane can never add up to give a null vector

4.4 State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:

(a) Adding any two scalars

(b) Adding a scalar to a vector of the same dimensions

(c) Multiplying any vector by any scalar

(d) Multiplying any two scalars

(e) Adding any two vectors

(f) Adding a component of a vector to the same vector

4.3 Pick out the only vector quantity in the following list: Temperature, pressure, impulse, time, power, total path length, energy

4.2 Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.