CLASS 11 · PHYSICS · CHAPTER 3

MOTION IN A PLANE

Simple worked solutions to all 22 end-of-chapter questions on vectors, projectile motion and circular motion.

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Chapter 3 Exercises

Textbook pages 47–49

Question 1

Classify these quantities as scalars or vectors: volume, mass, speed, acceleration, density, amount in moles, velocity, angular frequency, displacement and angular velocity.

Steps only

A scalar has magnitude only. A vector has magnitude and direction.

Answer

Scalars: volume, mass, speed, density, amount in moles and angular frequency. Vectors: acceleration, velocity, displacement and angular velocity.

Question 2

Pick the two scalars from force, angular momentum, work, electric current, linear momentum, electric field, average velocity, magnetic moment and relative velocity.

Steps only

Vectors have direction; scalars do not.

Answer

Work and electric current are the two scalars. The other listed quantities have directions.

Question 3

Pick the only vector from temperature, pressure, impulse, time, power, path length, energy, gravitational potential, friction coefficient and charge.

Steps only

A vector needs a direction as well as a size.

Answer

Impulse is the only vector. It has the direction of the change in momentum.

Question 4

Say whether each operation is meaningful, and why: (a) adding two scalars, (b) adding a scalar to a vector, (c) multiplying a vector by a scalar, (d) multiplying two scalars, (e) adding two vectors, (f) adding a vector component to that vector.

Steps only

Check that the quantities have compatible units and the same mathematical type. Vector addition also follows the head-to-tail rule.

Answer

(a) Yes, when the scalars are same-kind quantities with matching units. (b) No: scalar and vector types differ. (c) Yes; the scalar changes vector magnitude and a negative scalar reverses direction. (d) Yes. (e) Yes, for vectors of the same kind. (f) Yes if “component” means the component vector along an axis; add it as a vector. A scalar component by itself cannot be added to a vector.

Question 5

For each statement, say true or false: (a) magnitude of a vector is scalar; (b) every vector component is scalar; (c) path length always equals displacement magnitude; (d) average speed is at least the magnitude of average velocity; (e) three non-coplanar vectors can sum to zero.

Steps only

Use definitions: magnitude is a number; path counts the route; displacement joins start to finish; speed and velocity averages use path length and displacement respectively.

Answer

(a) True. (b) True for scalar components along axes. (c) False; equality holds only for a straight path without reversing. (d) True, because path length ≥ displacement magnitude. (e) True; if three vectors sum to zero, the third is the negative of the sum of the first two, so all lie in one plane.

Question 6

Show that (a) |a+b|≤|a|+|b|, (b) |a+b|≥||a|−|b||, (c) |a−b|≤|a|+|b|, (d) |a−b|≥||a|−|b||. When does equality hold?

Steps only

The triangle rule says one side of a triangle cannot be longer than the other two sides together. Apply it to vectors a and b, and then interchange or reverse a vector to get the lower bounds.

Answer

(a) Equality when a and b point in the same direction. (b) Equality when they point in opposite directions. (c) Equality when a and b point in opposite directions. (d) Equality when they point in the same direction.

Question 7

Given a+b+c+d=0, which statements must hold? (a) all four vectors are zero; (b) |a+c|=|b+d|; (c) |a| can never exceed |b|+|c|+|d|; (d) b+c lies in the plane of a and d when they are not parallel, and on their line when they are parallel.

Steps only

Rearrange the equation: a+c=−(b+d), and a=−(b+c+d). Also b+c=−(a+d).

Answer

(a) False; non-zero vectors may cancel. (b) True: opposite vectors have equal magnitudes. (c) True by the triangle inequality. (d) True because b+c=−(a+d), which lies in the plane (or line) containing a and d.

Question 8

Three skaters travel from P to the diametrically opposite point Q on a circular rink of radius 200 m by different paths shown in the figure. Find each displacement magnitude and say when it equals path length.

Steps only

Displacement depends only on start and finish, not on the route. P and Q are opposite ends of a diameter.

Answer

Each skater’s displacement magnitude is the diameter, 2×200=400 m. It equals path length only for the skater whose route is the straight diameter.

Question 9

A cyclist starts at the centre of a circular park of radius 1 km, rides to its edge, follows a semicircle to the opposite edge, then returns to the centre. The trip takes 10 min. Find (a) displacement, (b) average velocity and (c) average speed.

Steps only

The start and finish are both the centre, so net displacement is zero. The path is one radius out, a half-circumference, then one radius back: 1+π+1 km.

Answer

(a) 0 km. (b) 0/10 min=0. (c) Distance=(2+π) km≈5.14 km; time=1/6 h. Average speed≈5.14÷(1/6)=30.8 km/h.

Question 10

A motorist travels 500 m, then turns left by 60°; this repeats at every 500 m. Find displacement at the 3rd, 6th and 8th turns, and compare with path length.

Steps only

Each new leg points 60° left of the previous one. Six equal legs make a closed regular hexagon. At the 3rd turn the displacement is 1000 m; at the 6th it is zero; the 8th is equivalent to two legs from the start.

Answer

3rd: displacement magnitude 1000 m, path 1500 m. 6th: displacement 0, path 3000 m. 8th: displacement √3×500≈866 m, path 4000 m.

Question 11

A hotel is 10 km from a station along a straight road. A taxi takes a 23 km route and arrives in 28 min. Find (a) average speed and (b) magnitude of average velocity. Are they equal?

Steps only

Convert 28 min to 28/60 h. Average speed uses total path; average velocity magnitude uses straight displacement.

Answer

(a) 23÷(28/60)=49.3 km/h. (b) 10÷(28/60)=21.4 km/h. They are not equal because the taxi’s route was longer than the station-to-hotel displacement.

Question 12

A ball is thrown at 40 m/s in a hall with a 25 m ceiling. What is the greatest horizontal distance it can travel without hitting the ceiling? Take g=9.8 m/s².

Steps only

The best launch angle under this height limit makes the projectile just reach 25 m at its highest point. H=u²sin²θ/(2g), so sinθ=√(2gH)/u=√490/40≈0.553; θ≈33.5°.

Answer

Range R=u²sin(2θ)/g≈(1600/9.8)sin(67°)≈150 m.

Question 13

A cricketer can throw a ball a maximum horizontal distance of 100 m. How high can the same ball be thrown vertically?

Steps only

For a given launch speed, maximum range is Rmax=u²/g. A vertical throw reaches height H=u²/(2g).

Answer

H=Rmax/2=50 m.

Question 14

A stone on an 80 cm string makes 14 revolutions in 25 s at constant speed. Find the acceleration magnitude and direction.

Steps only

Radius r=0.80 m. Frequency f=14/25=0.56 s⁻¹, so angular speed ω=2πf≈3.52 rad/s. In uniform circular motion a=ω²r, directed inward.

Answer

a≈(3.52)²(0.80)=9.9 m/s², towards the centre of the circle.

Question 15

An aircraft flies in a horizontal circle of radius 1.00 km at 900 km/h. Compare its centripetal acceleration with g=9.8 m/s².

Steps only

Convert speed: 900 km/h=250 m/s; radius=1000 m. Use ac=v²/r.

Answer

ac=250²/1000=62.5 m/s². Compared with g: 62.5/9.8≈6.4. Its centripetal acceleration is about 6.4g.

Question 16

State whether each is true or false: (a) acceleration in circular motion always points to the centre; (b) velocity is tangent to the path; (c) average acceleration over one uniform circular cycle is zero.

Steps only

For non-uniform circular motion, acceleration can have both inward and tangential parts. Velocity is tangent to any smooth path. Over a full cycle in uniform circular motion, velocity returns to its starting value.

Answer

(a) False in general; it is purely inward only at constant speed. (b) True. (c) True: net change in velocity over a complete cycle is zero, so average acceleration is zero.

Question 17

A particle’s position is r=(3.0t î−2.0t² ĵ+4.0 k̂) m. Find (a) v and a, and (b) velocity magnitude and direction at t=2.0 s.

Steps only

Differentiate each position component once for velocity and again for acceleration. At t=2, substitute into v.

Answer

v=(3.0 î−4.0t ĵ) m/s; a=−4.0 ĵ m/s². At 2 s, v=(3 î−8 ĵ) m/s; |v|=√(3²+8²)=√73≈8.54 m/s. Direction=tan⁻¹(8/3)≈69.4° below +x.

Question 18

A particle starts at the origin with v₀=10 ĵ m/s and a=(8 î+2 ĵ) m/s². Find (a) when x=16 m and y then, and (b) its speed then.

Steps only

Use x=v₀x t+½axt²=4t². Setting x=16 gives t=2 s. For y, y=10t+½(2)t². Velocity at that time is v=v₀+at.

Answer

At t=2 s, y=20+4=24 m. v=16 î+14 ĵ m/s, so speed=√(16²+14²)=√452≈21.3 m/s.

Question 19

Find magnitude and direction of î+ĵ and î−ĵ. Then find the components of A=2î+3ĵ along these two directions.

Steps only

Each sum has perpendicular unit components, so its magnitude is √(1²+1²)=√2. Their directions are +45° and −45°. Unit vectors along them are (î+ĵ)/√2 and (î−ĵ)/√2.

Answer

Scalar projections of A are A·n₁=5/√2 and A·n₂=−1/√2. As vector components these are (5/2)(î+ĵ) and (−1/2)(î−ĵ).

Question 20

For arbitrary motion, which statements are always true? (a) average velocity equals the mean of endpoint velocities; (b) average velocity equals displacement/time; (c) v(t)=v(0)+at; (d) r(t)=r(0)+v(0)t+½at²; (e) average acceleration equals velocity change/time.

Steps only

Definitions (b) and (e) work for any motion. Formulae (a), (c) and (d) need constant acceleration.

Answer

(a) False in general. (b) True. (c) False in general; true for constant acceleration. (d) False in general; true for constant acceleration. (e) True by definition.

Question 21

For each claim about scalars, say true or false and give a reason or example: (a) a scalar is conserved; (b) a scalar cannot be negative; (c) a scalar must be dimensionless; (d) a scalar cannot vary by position; (e) a scalar has the same value for observers with differently oriented axes.

Steps only

“Scalar” means the quantity has magnitude without direction. It does not by itself say whether the quantity is conserved, positive, dimensionless or uniform in space.

Answer

(a) False: energy may change form in a chosen system. (b) False: temperature in °C may be negative. (c) False: mass has units. (d) False: temperature can vary from place to place. (e) True: changing axis orientation changes vector components, not a scalar value.

Question 22

An aircraft flies 3400 m above a ground observer. The angle between its apparent positions 10.0 s apart is 30°. Estimate its speed, assuming it flies horizontally at constant height.

Steps only

Using the textbook’s right-triangle geometry, the horizontal distance for the stated angular change is h tan30°=3400/√3≈1963 m. Divide distance by elapsed time.

Answer

Speed≈1963/10≈196 m/s (about 707 km/h).