Question 1

A car containing explosives goes over a ramp of 45º angle and initial velocity of 20 m/sec. After 2 seconds, explosion occurs. What can be said about the trajectory of the centre of mass of the car after explosion occurs as compared to trajectory of the car without explosion.

Accepted Answer

No change in the horizontal range — [{"id": "07a4b162-a78d-917e-25d6-26b10621dfe2", "type": "html", "value": " Even though each piece of car falls apart and move away from each other rapidly, the net acceleration acting on the centre of mass of car (pieces of car) will be zero. Internal forces(explosion) do not alter the horizontal range of the centre of mass of car. "}]

Question 2

A body consisting of four equal masses is subjected to pure rotational motion in space at a constant angular velocity. Each mass is released at 0º , 45º, 90º and 135º. Which of the following statement is true with respect to the centre of mass of the body after release of all four masses -

Accepted Answer

Centre of mass will be stationary and away from the centre of rotation. — [{"id": "dc155c17-10ae-b85d-d85b-5e68e63f4bcc", "type": "html", "value": " Option 1 is false as the mass thrown at 0º would have covered greater distance than other three masses, the mass thrown at 45º would have covered greater distance than other two masses and so on; which means that the centre of mass will move away from the centre of rotation. Option 3 and 4 are false because net velocity of the masses will be zero. "}]

Question 3

1 State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.

Accepted Answer

Volume - Scalar (has magnitude only)
Mass - Scalar
Speed - Scalar
Acceleration - Vector (has magnitude and direction)
Density - Scalar
Number of moles - Scalar
Velocity - Vector
Angular frequency - Scalar
Displacement - Vector
Angular velocity - Vector — Scalars have only magnitude, vectors have both magnitude and direction. Quantities like volume, mass, speed, density, number of moles, and angular frequency are scalars. Acceleration, velocity, displacement, and angular velocity have direction and magnitude, so they are vectors.

Question 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.

Accepted Answer

The two scalar quantities are: work and current.
Explanation: Work is scalar because it has magnitude only. Current is scalar as it is a measure of charge flow rate without direction. Force, angular momentum, linear momentum, electric field, average velocity, magnetic moment, and relative velocity are vectors. — Scalars have magnitude only; vectors have magnitude and direction. Work and current are scalars; the rest are vectors.

Question 5

3 Pick out the only vector quantity in the following list: Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

Accepted Answer

The only vector quantity is impulse.
Explanation: Impulse is a vector because it is the product of force (vector) and time (scalar), thus has direction and magnitude. Temperature, pressure, time, power, total path length, energy, gravitational potential, coefficient of friction, and charge are scalars. — Impulse is vector; all other quantities listed are scalars.

Question 6

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.

Accepted Answer

(a) Adding any two scalars: Meaningful. Scalars have magnitude only and can be added algebraically.

(b) Adding a scalar to a vector of the same dimensions: Not meaningful. Scalars and vectors are different types; addition requires same type.

(c) Multiplying any vector by any scalar: Meaningful. Scalar multiplication of vector changes magnitude but not direction.

(d) Multiplying any two scalars: Meaningful. Scalars multiply algebraically.

(e) Adding any two vectors: Meaningful. Vector addition is defined.

(f) Adding a component of a vector to the same vector: Not meaningful. A component is scalar; vector plus scalar is not defined. — Operations must be between compatible types. Scalars can be added or multiplied with scalars. Vectors can be added or multiplied by scalars. Adding scalar to vector or component to vector is not defined.

Question 7

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.

Accepted Answer

(a) True. Magnitude is scalar quantity.
(b) True. Components are projections along axes and are scalars.
(c) False. Total path length is generally greater than or equal to magnitude of displacement.
(d) True. Average speed ≥ magnitude of average velocity.
(e) False. Three vectors not in a plane can add to zero if arranged properly in 3D space. — Magnitude and components of vectors are scalars. Path length ≥ displacement magnitude. Average speed is path length/time, average velocity is displacement/time. Three non-coplanar vectors can sum to zero in 3D.

Question 8

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?

Accepted Answer

These are triangle inequalities for vectors.

(a) |a + b| ≤ |a| + |b|: The magnitude of the sum of two vectors is less than or equal to the sum of their magnitudes.
Equality holds when vectors a and b are in the same direction.

Proof: Using the law of cosines,
|a + b|^2 = |a|^2 + |b|^2 + 2|a||b|cosθ ≤ (|a| + |b|)^2

(b) |a + b| ≥ ||a| - |b||: The magnitude of the sum is greater than or equal to the difference of magnitudes.
Equality holds when vectors a and b are in opposite directions.

(c) |a - b| ≤ |a| + |b|: Similar to (a), since a - b = a + (-b), triangle inequality applies.

(d) |a - b| ≥ ||a| - |b||: Similar to (b), equality when a and b are collinear.

Summary: Equality holds when vectors are collinear (either same or opposite direction). — These inequalities follow from the triangle inequality for vectors and the law of cosines. Equality occurs when vectors are parallel or anti-parallel.

Question 9

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?

Accepted Answer

(a) False. Vectors can sum to zero without each being null.
(b) True. From a + b + c + d = 0, rearranged as a + c = -(b + d), magnitudes equal.
(c) True. By triangle inequality, |a| ≤ |b| + |c| + |d|.
(d) True. For the sum to be zero, b + c must lie in the plane or line defined by a and d depending on their collinearity. — Vector addition properties and geometric interpretation of vector sums explain these statements.

Question 10

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. 3.19. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of path skate?

Accepted Answer

Displacement vector magnitude for each girl is the straight line distance from P to Q, which is the diameter of the circle = 2 × radius = 2 × 200 m = 400 m.

The actual path length differs:
- For the girl skating along the semicircular path, path length = half the circumference = π × radius = π × 200 ≈ 628 m.
- For the girl skating along the straight line (if any), path length = displacement = 400 m.

Therefore, the magnitude of displacement is 400 m for all, but only the girl skating along the straight line path has displacement equal to actual path length. — Displacement is the shortest distance between initial and final points, here the diameter. Path length depends on actual route taken.

Question 11

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. 3.20. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist?

Accepted Answer

(a) Net displacement = 0 m (since start and end points are the same at centre O).

(b) Average velocity = displacement / time = 0 / (10 × 60) = 0 m/s.

(c) Average speed = total path length / time.
Total path length = OP + PQ + QO
OP = radius = 1 km = 1000 m
PQ = half circumference = π × radius = π × 1000 ≈ 3141.6 m
QO = radius = 1000 m
Total path length = 1000 + 3141.6 + 1000 = 5141.6 m
Time = 10 min = 600 s
Average speed = 5141.6 / 600 ≈ 8.57 m/s — Displacement is vector from initial to final point; here zero. Average velocity is displacement/time. Average speed is total distance/time.

Question 12

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.

Accepted Answer

The motorist moves 500 m straight, then turns left 60° after each 500 m.

At each turn, the displacement vector is the vector sum of the segments.

Third turn:
Number of segments = 3
Each segment length = 500 m
Total path length = 3 × 500 = 1500 m
Displacement magnitude can be found by vector addition of three vectors each 500 m, with 60° turn between them.
Using vector addition or cosine law, displacement ≈ 866 m.

Sixth turn:
Total path length = 6 × 500 = 3000 m
Displacement vector sum of six vectors each 500 m, turning 60° each time.
After 6 turns, the motorist completes a closed hexagon, so displacement = 0.

Eighth turn:
Total path length = 8 × 500 = 4000 m
Displacement vector sum of eight vectors each 500 m, turning 60° each time.
After 6 turns, displacement is zero; next two segments add vectorially.
Calculate displacement of last two segments: 2 × 500 m with 60° turn between them.
Displacement magnitude ≈ 866 m.

Comparison:
Displacement magnitude ≤ total path length always.
At 6th turn, displacement = 0 but path length = 3000 m.
At 3rd and 8th turns, displacement is less than path length. — Displacement is vector sum of segments; path length is scalar sum. Closed polygon after 6 turns gives zero displacement.

Question 13

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?

Accepted Answer

(a) Average speed = total distance / time = 23 km / (28/60) h = 23 / 0.4667 ≈ 49.3 km/h

(b) Magnitude of average velocity = displacement / time = 10 km / 0.4667 h ≈ 21.43 km/h

Are they equal? No, average speed is greater than magnitude of average velocity because path length > displacement. — Average speed depends on total distance traveled; average velocity depends on displacement. Since path is longer than straight line, speed > velocity magnitude.

Question 14

12 The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m/s can go without hitting the ceiling of the hall?

Accepted Answer

Let the ball be thrown at an angle θ with speed u = 40 m/s.
Maximum height H = (u^2 sin^2 θ) / (2g) ≤ 25 m

We want to maximize horizontal range R = (u^2 sin 2θ) / g subject to H ≤ 25 m.

From H ≤ 25,
(u^2 sin^2 θ) / (2g) ≤ 25
sin^2 θ ≤ (2g × 25) / u^2 = (2 × 9.8 × 25) / 1600 = 490 / 1600 = 0.30625
sin θ ≤ 0.5534

Horizontal range R = (u^2 sin 2θ) / g = (u^2 × 2 sin θ cos θ) / g
= (2 u^2 sin θ cos θ) / g

For sin θ = 0.5534, cos θ = sqrt(1 - 0.5534^2) = 0.8329

R = (2 × 1600 × 0.5534 × 0.8329) / 9.8 ≈ (2 × 1600 × 0.4609) / 9.8 ≈ (1475) / 9.8 ≈ 150.5 m

Therefore, maximum horizontal distance without hitting ceiling is approximately 150.5 m. — Maximum height constraint limits the angle of projection, which in turn limits horizontal range.

Question 15

13 A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball?

Accepted Answer

Maximum horizontal range R = 100 m.
For projectile motion, R = (u^2) / g when angle = 45°.

So, u^2 = R × g = 100 × 9.8 = 980 m^2/s^2

Maximum height H = (u^2) / (2g) = 980 / (2 × 9.8) = 980 / 19.6 = 50 m

Therefore, the cricketer can throw the ball up to 50 m high. — Maximum range occurs at 45°, and maximum height is half the range times g.

Motion in a Straight Line

Motion in a Straight Line — Study Notes

3.1 Introduction

3.2 Scalars and vectors

3.3 Multiplication of vectors by real numbers

Practice Questions — Motion in a Straight Line

All 7 Chapters in Physics Part-I