1 Marks Question

Question 11 Mark

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

A scalar quantity is one that

Has the same value for observers with different orientations of axes.

Answer

True.
Explanation:
The value of a scalar does not vary for observers with different orientations of axes.

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Question 21 Mark

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

A scalar quantity is one that

Does not vary from one point to another in space.

Answer

False.
Explanation:
A scalar quantity such as gravitational potential can vary from one point to another in space.

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Question 31 Mark

Read each statement below carefully and state, with reasons, 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.

Answer

False.
Explanation:
The net acceleration of a particle is towards the centre only in case of a uniform circular motion.

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Question 41 Mark

A passenger arriving in a new town wishes to go from the station to a hotel located 10km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23km 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?

Answer

Total distance travelled = 23km
Total time taken = 28 min $=\frac{28}{60}\text{h}$
$\therefore$ Average speed of the taxi $=\frac{\text{Total distance travelled}}{\text{Total time taken}}$
$=\frac{23}{\frac{28}{60}}=49.29\text{km/h}$
Distance between the hotel and the station = 10km = Displacement of the car
$\therefore$ Average velocity $=\frac{10}{\frac{28}{60}}=21.43\text{km/h}$
Therefore, the two physical quantities (average speed and average velocity) are not equal.

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Question 51 Mark

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

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?

Answer

Correct.

Explanation:

a + b + c + d = 0,

a + (b + c) + d = 0,

The resultant sum of the three vectors a, (b + c), and d can be zero only if (b + c) lie in a plane containing a and d, assuming that these three vectors are represented by the three sides of a triangle.

If a and d are collinear, then it implies that the vector (b + c) is in the line of a and d. This implication holds only then the vector sum of all the vectors will be zero.

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Question 61 Mark

For any arbitrary motion in space, which of the following relations are true:
$\text{v}(\text{t})=\text{v}(0)+\text{a t}$
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

Answer

False.
Explanation:
The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.

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Question 71 Mark

For any arbitrary motion in space, which of the following relations are true:
$\text{v}_\text{average}=\frac{1}{2}(\text{v}(\text{t}_1)+\text{v}(\text{t}_2))$
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

Answer

False.
Explanation:
It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.

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Question 81 Mark

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

Answer

Radius of the loop, r = 1km = 1000m
Speed of the aircraft, $\text{v}=900\text{km/h}=900\times\frac{5}{18}=250\text{m/s}$
Centripetal acceleration, $\text{a}_\text{c}=\frac{\text{v}^2}{\text{r}}$
$=\frac{250^2}{1000}=62.5\text{m/s}^2$
Acceleration due to gravity, g = 9.8m/s²
$\frac{\text{a}_\text{c}}{\text{g}}=\frac{62.5}{9.8}=6.38$
$\text{a}_\text{c}=6.38\text{g}$

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Question 91 Mark

Read each statement below carefully and state with reasons, 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.

Answer

True.
Explanation:
It is because of the fact that the total path length is always greater than or equal to the magnitude of displacement of a particle.

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Question 101 Mark

For any arbitrary motion in space, which of the following relations are true:
$\text{v}(\text{t})=\text{r}(0)+\text{v}(0)\text{t}+\frac{1}{2}\text{a t}^2$
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

Answer

False.
Explanation:
The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.

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Question 111 Mark

Read each statement below carefully and state with reasons, if it is true or false:
The total path length is always equal to the magnitude of the displacement vector of a particle.

Answer

False.
Explanation:
Total path length is a scalar quantity, whereas displacement is a vector quantity. Hence, the total path length is always greater than the magnitude of displacement. It becomes equal to the magnitude of displacement only when a particle is moving in a straight line.

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Question 121 Mark

Read each statement below carefully and state with reasons, if it is true or false:
Each component of a vector is always a scalar,

Answer

False.
Explanation:
Each component of a vector is also a vector.

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Question 131 Mark

For any arbitrary motion in space, which of the following relations are true:
$\text{v}_{\text{average}}=\frac{[\text{r}(\text{t}_2)-\text{r}(\text{t}_1)]}{(\text{t}_2-\text{t}_1)}$
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

Answer

True.
Explanation:
The arbitrary motion of the particle can be represented by ythis equation.

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Question 141 Mark

Read each statement below carefully and state with reasons, if it is true or false:
The magnitude of a vector is always a scalar,

Answer

True.
Explanation:
The magnitude of a vector is a number. Hence, it is a scalar.

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Question 151 Mark

Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
Can never take negative values.

Answer

False.
Explanation:
Despite being a scalar quantity, temperature can take negative values.

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Question 161 Mark

For any arbitrary motion in space, which of the following relations are true:
$\text{a}_{\text{average}}=\frac{[\text{v}(\text{t}_2)-\text{v}(\text{t}_1)]}{(\text{t}_2-\text{t}_1)}$
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

Answer

True.
Explanation:
The arbitrary motion of the particle can be represented by this equation.

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Question 171 Mark

Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
Must be dimensionless.

Answer

False.
Explanation:
Total path length is a scalar quantity. Yet it has the dimension of length.

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Question 181 Mark

Read each statement below carefully and state with reasons, if it is true or false:
Three vectors not lying in a plane can never add up to give a null vector.

Answer

True.
Explanation:
Three vectors, which do not lie in a plane, cannot be represented by the sides of a triangle taken in the same order.

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Question 191 Mark

Read each statement below carefully and state, with reasons, if it is true or false:
The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.

Answer

True.

Explanation:

The direction of acceleration vector in a uniform circular motion is directed towards the centre of circular path. It is constantly changing with time. The resultant of all these vectors will be zero vector.

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Question 201 Mark

Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
Is conserved in a process.

Answer

False.
Explanation:
Despite being a scalar quantity, energy is not conserved in inelastic collisions.

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Question 211 Mark

Read each statement below carefully and state, with reasons, 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.

Answer

True.

Explanation:

because while leaving the circular path, the particle moves tangentially to the circular path.

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Question 221 Mark

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

adding any two scalars.
adding a scalar to a vector of the same dimensions.
multiplying any vector by any scalar.
multiplying any two scalars.
adding any two vectors.
adding a component of a vector to the same vector.

Answer

Yes, addition of two scalar quantities is meaningful only if they both represent the same physical quantity.
No, addition of a vector quantity with a scalar quantity is not meaningful.
Yes, scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.
Yes, scalar, irrespective of the physical quantity it represents, can be multiplied with another scalar having the same or different dimensions.
Yes, addition of two vector quantities is meaningful only if they both represent the same physical quantity.
Yes, component of a vector can be added to the same vector as they both have the same dimensions.

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Question 231 Mark

Galileo, in his book Two new sciences, stated that “for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal”. Prove this statement.

Answer

For a projectile launched with velocity $v _{ o }$ at an angle $\theta_{ o }$, the range is given by
$
R=\frac{v_0^2 \sin 2 \theta_0}{g}
$
Now, for angles, $(45+\alpha)$ and $(45-\alpha), 2 \theta_0$ is $(90+2 \alpha)$ and $(90-2 \alpha)$, respectively. The values of $\sin (90+2 \alpha)$ and $\sin (90-2 \alpha)$ are the same, equal to that of $\cos 2 \alpha$. Therefore, ranges are equal for elevations which exceed or fall short of 45 by equal amounts $\alpha$.

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