NCERT solutions for Physics [English] Class 11 chapter 3 - Motion in a Plane [Latest edition]

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.

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.

Pick out the only vector quantity in the following list:

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

1. adding any two scalars,
2. adding a scalar to a vector of the same dimensions,
3. multiplying any vector by any scalar,
4. multiplying any two scalars,
5. adding any two vectors,
6. adding a component of a vector to the same vector.

Read the statement below carefully and state with reason, if it is true or false:

The magnitude of a vector is always a scalar.

Read the statement below carefully and state with reason, if it is true or false:

Each component of a vector is always a scalar.

Read the statement below carefully and state with reason, if it is true or false:

The total path length is always equal to the magnitude of the displacement vector of a particle. 

Read the statement below carefully and state with reason, if it is true or false: 

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.

Read the statement below carefully and state with reason, if it is true or false:

Three vectors not lying in a plane can never add up to give a null vector.

Establish the following vector inequalities geometrically or otherwise:

1. | a+b | ≤ |a| + |b|
2. |a + b| ≥ | |a| - |b| |
3. |a - b| ≤ |a| + |b|
4. |a-b| ≥ | |a| - |b| |

When does the equality sign above apply?

Given a + b + c + d = 0, state whether the following statement is correct or incorrect:

a, b, c, and d must each be a null vector.

Given a + b + c + d = 0, state whether the following statement is correct or incorrect: 

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

Given a + b + c + d = 0, state whether the following statement is correct or incorrect: 

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

Given a + b + c + d = 0, state whether the following statement is correct or incorrect:

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.

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

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

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.

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.

1. What is the average speed of the taxi,
2. the magnitude of average velocity? Are the two equal?

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^–1 can go without hitting the ceiling of the hall?

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?

A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone?

An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.

Read the statement below carefully and state, with reason, if it is true or false:

The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre.

Read the statement below carefully and state, with reason, if it is true or false:

The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.

Read the statement below carefully and state, with reason, if it is true or false:

The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.

The position of a particle is given by

`r = 3.0t  hati − 2.0t 2  hatj + 4.0  hatk m`

Where t is in seconds and the coefficients have the proper units for r to be in metres.

1. Find the v and a of the particle?
2. What is the magnitude and direction of velocity of the particle at t = 2.0 s?

A particle starts from the origin at t = 0 s with a velocity of 10.0 `hatj "m/s"` and moves in the x-y plane with a constant acceleration of `(8.0 hati + 2.0 hatj) ms^(-2)`.

1. At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
2. What is the speed of the particle at the time?

`hati  "and"  hatj` are unit vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors `hati+hatj` and `hati-hatj` ? What are the components of a vector `A = 2hati + 3hatj` along the directions of `hati + hatj` and `hati - hatj` ? [You may use graphical method]

For any arbitrary motion in space, state whether the following statement is true:

`"V"_"average"` = `(1/2)("v"("t"_1) + "v"("t"_2))`

For any arbitrary motion in space, state whether the following statement is true:

`"V"_"average"` = [r(t₂) - r(t₁) ] /(t₂ – t₁)

For any arbitrary motion in space, state whether the following statement is true:

v (t) = v (0) + a t

(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

For any arbitrary motion in space, state whether the following statement is true:

`"r"("t") = "r"(0) + "v"(0)"t"  + 1/2  "a"  "t"^2 `

(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

For any arbitrary motion in space, state whether the following statement is true:

`"a"_"average"=["v"("t"_2) - "v"("t"_1)]/("t"_2 - "t"_1)`

(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

Read the statement below carefully and state, with reason and example, if it is true or false:

A scalar quantity is one that is conserved in a process.

Read the statement below carefully and state, with reason and example, if it is true or false:

A scalar quantity is one that can never take negative values.

Read the statement below carefully and state, with reason and example, if it is true or false:

A scalar quantity is one that must be dimensionless.

Read the statement below carefully and state, with reason and example, if it is true or false:

A scalar quantity is one that  does not vary from one point to another in space.

An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s a part is 30°, what is the speed of the aircraft?