Motion In A Plane - NEET/JEE/KEAM

1. The vectors A and B are such that, |A+B| = |A-B|. The angle between two vectors is,

60°

75°

45°

90°

If |A+B| = |A-B|, it means that the magnitudes of the sum of vectors A and B and the difference of vectors A and B are equal. This implies that the vectors A and B have the same magnitude. In geometric terms, this means that the vectors A and B lie on the same circle centered at the origin. The angle between two vectors on a circle centered at the origin is always 90°. Therefore, the angle between vectors A and B is 90°.

Explanation

If |A+B| = |A-B|, it means that the magnitudes of the sum of vectors A and B and the difference of vectors A and B are equal. This implies that the vectors A and B have the same magnitude. In geometric terms, this means that the vectors A and B lie on the same circle centered at the origin. The angle between two vectors on a circle centered at the origin is always 90°. Therefore, the angle between vectors A and B is 90°.

2. Which of the following is not a vector quantity?

Speed

Velocity

Torque

Displacement

Speed is not a vector quantity because it only represents the magnitude of motion without any direction. It is a scalar quantity that can be measured in terms of distance covered per unit time. On the other hand, velocity, torque, and displacement are all vector quantities as they involve both magnitude and direction. Velocity represents the rate of change of displacement, torque represents the rotational force acting on an object, and displacement represents the change in position of an object in a particular direction.

Explanation

Speed is not a vector quantity because it only represents the magnitude of motion without any direction. It is a scalar quantity that can be measured in terms of distance covered per unit time. On the other hand, velocity, torque, and displacement are all vector quantities as they involve both magnitude and direction. Velocity represents the rate of change of displacement, torque represents the rotational force acting on an object, and displacement represents the change in position of an object in a particular direction.

3. If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is:

0°

90°

45°

180°

If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, it implies that the vectors are perpendicular to each other. This is because the magnitude of the sum of two vectors is equal to the magnitude of the diagonal of a parallelogram formed by the vectors, while the magnitude of the difference of the two vectors is equal to the magnitude of the base of the parallelogram. In a parallelogram, the diagonal and the base are perpendicular to each other. Therefore, the angle between the vectors is 90°.

Explanation

If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, it implies that the vectors are perpendicular to each other. This is because the magnitude of the sum of two vectors is equal to the magnitude of the diagonal of a parallelogram formed by the vectors, while the magnitude of the difference of the two vectors is equal to the magnitude of the base of the parallelogram. In a parallelogram, the diagonal and the base are perpendicular to each other. Therefore, the angle between the vectors is 90°.

4. A particle moves in a plane with constant acceleration in a direction different from the initial velocity. The path of the particle is,

An ellipse

A parabola

An arc of a circle

A straight line

When a particle moves in a plane with constant acceleration in a direction different from the initial velocity, the path of the particle is a parabola. This is because the combination of the initial velocity and the constant acceleration causes the particle to follow a curved trajectory, which is characteristic of a parabolic path.

Explanation

When a particle moves in a plane with constant acceleration in a direction different from the initial velocity, the path of the particle is a parabola. This is because the combination of the initial velocity and the constant acceleration causes the particle to follow a curved trajectory, which is characteristic of a parabolic path.

5. Which of the following is not a vector quantity?

Displacement

Electric field

Work

Acceleration

Work is not a vector quantity because it is a scalar quantity. Scalar quantities only have magnitude and no direction. Work is defined as the product of force and displacement, and it only represents the amount of energy transferred or expended in a particular direction. It does not have a specific direction associated with it, unlike vector quantities such as displacement, electric field, and acceleration, which have both magnitude and direction.

Explanation

Work is not a vector quantity because it is a scalar quantity. Scalar quantities only have magnitude and no direction. Work is defined as the product of force and displacement, and it only represents the amount of energy transferred or expended in a particular direction. It does not have a specific direction associated with it, unlike vector quantities such as displacement, electric field, and acceleration, which have both magnitude and direction.

6. The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

Cannot be predicted

Are equal to each other

Are equal to each other in magnitude

Are not equal to each other in magnitude

If the vector sum of two forces is perpendicular to their vector differences, it means that the two forces are acting at right angles to each other. In this case, the magnitude of the forces must be equal in order for their vector sum to be perpendicular to their vector difference. Therefore, the correct answer is that the forces are equal to each other in magnitude.

Explanation

If the vector sum of two forces is perpendicular to their vector differences, it means that the two forces are acting at right angles to each other. In this case, the magnitude of the forces must be equal in order for their vector sum to be perpendicular to their vector difference. Therefore, the correct answer is that the forces are equal to each other in magnitude.

7. Two bodies of same mass are projected with the same velocity at an angle 30° and 60° respectively. The ratio of their horizontal ranges will be,

1:1

1:2

1:3

1:6

When two bodies of the same mass are projected with the same velocity at different angles, the ratio of their horizontal ranges will be 1:1. This is because the horizontal component of the velocity is the same for both bodies, regardless of the angle at which they are projected. Therefore, the horizontal distances covered by the two bodies will be equal.

Explanation

When two bodies of the same mass are projected with the same velocity at different angles, the ratio of their horizontal ranges will be 1:1. This is because the horizontal component of the velocity is the same for both bodies, regardless of the angle at which they are projected. Therefore, the horizontal distances covered by the two bodies will be equal.

8. A car runs at a constant speed on a track of radius 100m, taking 62.8 seconds in every circular loop. The average velocity and average speed of each circular loop respectively, is

0, 10 m/s

10 m/s, 10 m/s

10m/s, 0

0, 0

The average velocity of each circular loop is 0 because velocity is a vector quantity that includes both magnitude and direction. Since the car is moving in a circular path, its velocity is constantly changing direction, resulting in an average velocity of 0.

The average speed of each circular loop is 10 m/s because speed is a scalar quantity that only considers the magnitude of the velocity. The car is running at a constant speed, so the magnitude of its velocity remains the same throughout each circular loop, resulting in an average speed of 10 m/s.

Explanation

The average velocity of each circular loop is 0 because velocity is a vector quantity that includes both magnitude and direction. Since the car is moving in a circular path, its velocity is constantly changing direction, resulting in an average velocity of 0.

The average speed of each circular loop is 10 m/s because speed is a scalar quantity that only considers the magnitude of the velocity. The car is running at a constant speed, so the magnitude of its velocity remains the same throughout each circular loop, resulting in an average speed of 10 m/s.

9. Vectors A, B, C are such thatA.B = 0 and A.C= 0. Then the vector parallel to A is,

B and C

A × B

B + C

B × C

If A.B = 0 and A.C = 0, it means that vector A is perpendicular to both vectors B and C. Therefore, the vector parallel to A would be the cross product of vectors B and C, which is denoted as B × C.

Explanation

If A.B = 0 and A.C = 0, it means that vector A is perpendicular to both vectors B and C. Therefore, the vector parallel to A would be the cross product of vectors B and C, which is denoted as B × C.

10. The resultant of (Am0) will be equal to,

Zero

A

Zero vector

Unit vector

When any vector is multiplied by zero, the resultant vector is always the zero vector. This is because multiplying any vector by zero essentially cancels out its magnitude and direction, resulting in a vector with a magnitude of zero and no specific direction. Therefore, the correct answer is the zero vector.

Explanation

When any vector is multiplied by zero, the resultant vector is always the zero vector. This is because multiplying any vector by zero essentially cancels out its magnitude and direction, resulting in a vector with a magnitude of zero and no specific direction. Therefore, the correct answer is the zero vector.

11. If a vector 2i+3j+8k is perpendicular to the vector 4j-4i+ πk, then the value of π is,

1/2

-1/2

1

-1

If two vectors are perpendicular, their dot product is zero. By taking the dot product of the given vectors, we get (2)(-4) + (3)(4) + (8)(π) = 0. Simplifying this equation, we find that 8π - 8 = 0. Solving for π, we get π = -1/2. Therefore, the value of π is -1/2.

Explanation

If two vectors are perpendicular, their dot product is zero. By taking the dot product of the given vectors, we get (2)(-4) + (3)(4) + (8)(π) = 0. Simplifying this equation, we find that 8π - 8 = 0. Solving for π, we get π = -1/2. Therefore, the value of π is -1/2.

12. The angle between two vectors of magnitude 12 and 18 units when their resultant is 24 units is,

63°51'

75°52'

82°31'

89°16'

The angle between two vectors can be found using the law of cosines. Let's assume the angle between the two vectors is θ. According to the law of cosines, the magnitude of the resultant vector is given by:

R^2 = A^2 + B^2 - 2ABcos(θ)

Substituting the given values, we have:

24^2 = 12^2 + 18^2 - 2(12)(18)cos(θ)

576 = 144 + 324 - 432cos(θ)

432cos(θ) = 468

cos(θ) = 468/432

θ = cos^(-1)(468/432)

θ ≈ 75.87 degrees

Converting to degrees and minutes, we get 75°52'. Therefore, the correct answer is 75°52'.

Explanation

The angle between two vectors can be found using the law of cosines. Let's assume the angle between the two vectors is θ. According to the law of cosines, the magnitude of the resultant vector is given by:

R^2 = A^2 + B^2 - 2ABcos(θ)

Substituting the given values, we have:

24^2 = 12^2 + 18^2 - 2(12)(18)cos(θ)

576 = 144 + 324 - 432cos(θ)

432cos(θ) = 468

cos(θ) = 468/432

θ = cos^(-1)(468/432)

θ ≈ 75.87 degrees

Converting to degrees and minutes, we get 75°52'. Therefore, the correct answer is 75°52'.

13. A ball of mass 0.25 kg attached to the end of a string of length 1.96m is moving in a horizontal circle. The string will break if the tension is more than 25N. What is the maximum speed with which the ball can be moved?

14 m/s

3 m/s

5 m/s

3.92 m/s

The maximum speed with which the ball can be moved is 14 m/s. This can be determined by using the centripetal force equation, which states that the centripetal force is equal to the mass of the object times its velocity squared divided by the radius of the circle. In this case, the tension in the string provides the centripetal force, so we can set it equal to the maximum tension of 25N. Rearranging the equation and solving for velocity, we find that the maximum speed is 14 m/s.

Explanation

The maximum speed with which the ball can be moved is 14 m/s. This can be determined by using the centripetal force equation, which states that the centripetal force is equal to the mass of the object times its velocity squared divided by the radius of the circle. In this case, the tension in the string provides the centripetal force, so we can set it equal to the maximum tension of 25N. Rearranging the equation and solving for velocity, we find that the maximum speed is 14 m/s.

14. When a body moves with a constant speed along a circle,

Its velocity remains constant

No force acts on it

No work is done on it

No acceleration is produced in it

When a body moves with a constant speed along a circle, no work is done on it because work is defined as the product of force and displacement, and in this case, there is no force acting on the body. The velocity remains constant because the speed is constant, but the direction of the velocity changes continuously due to the circular motion. Since there is no force acting on the body, there is no acceleration produced in it according to Newton's second law.

Explanation

When a body moves with a constant speed along a circle, no work is done on it because work is defined as the product of force and displacement, and in this case, there is no force acting on the body. The velocity remains constant because the speed is constant, but the direction of the velocity changes continuously due to the circular motion. Since there is no force acting on the body, there is no acceleration produced in it according to Newton's second law.

15. The angle between the two vectors A= 3i+ 4j+ 5k and B= 3i+ 4j + 5k will be

Zero

45°

90°

180°

The angle between two vectors can be found using the dot product formula: cosθ = (A · B) / (|A| |B|). In this case, both vectors A and B are the same, so the dot product will be equal to the product of their magnitudes: (A · B) = |A| |B|. Since the magnitudes of both vectors are equal, the dot product will be equal to the square of the magnitude: (A · B) = |A|^2. Plugging this into the formula, cosθ = |A|^2 / (|A| |B|) = |A| / |B|. Simplifying further, cosθ = 1, which means that the angle θ is 90°.

Explanation

The angle between two vectors can be found using the dot product formula: cosθ = (A · B) / (|A| |B|). In this case, both vectors A and B are the same, so the dot product will be equal to the product of their magnitudes: (A · B) = |A| |B|. Since the magnitudes of both vectors are equal, the dot product will be equal to the square of the magnitude: (A · B) = |A|^2. Plugging this into the formula, cosθ = |A|^2 / (|A| |B|) = |A| / |B|. Simplifying further, cosθ = 1, which means that the angle θ is 90°.