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Physics · MOTION IN A PLANE
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Physics · MOTION IN A PLANE
A motorboat is racing towards north at \( 25\text{ km/h} \) and the water current in that region is \( 10\text{ km/h} \) in the direction of \( 60^\circ \) east of south as shown in the vector diagram. The direction angle \( \phi \) of the resultant velocity vector relative to the north direction is approximately:
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A particle undergoes uniform circular motion in a path of radius \( R \). If its frequency of revolution is doubled while the radius of the circle is halved, the new centripetal acceleration becomes how many times its initial centripetal acceleration?
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A particle moves along a semi-circular path of radius \( R = 8\text{ m} \) from point A to point B. From point B, it turns and moves along a straight line segment of length \( 12\text{ m} \) perpendicular to the diameter AB. What is the magnitude of the total displacement of the particle from its initial position A?
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The position coordinates of a particle moving in the \( x\text{-}y \) plane are \( x(t) = 2t^3 - 3t^2 \) and \( y(t) = t^3 - 12t \) (where \( x, y \) are in metres and \( t \) in seconds). At what non-zero time \( t \) is the \( y \)-component of velocity equal to zero, and what is the speed of the particle at that instant?
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A particle thrown vertically upwards from ground reaches a maximum height \( H \). The ratio of the time taken by the particle to reach height \( \frac{3}{4}H \) during its upward motion to the total time of flight \( T_f \) is:
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Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) have equal magnitudes \( F \). If the magnitude of their resultant vector \( \mathbf{R} = \mathbf{A} + \mathbf{B} \) is also equal to \( F \), what is the angle between \( \mathbf{A} \) and \( \mathbf{B} \), and what is the magnitude of their vector difference \( |\mathbf{A} - \mathbf{B}| \)?
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If \( |\mathbf{A}| = 3 \), \( |\mathbf{B}| = 4 \), and the magnitude of their sum is \( |\mathbf{A} + \mathbf{B}| = 5 \), evaluate the magnitude of the vector linear combination \( |\mathbf{A} - 2\mathbf{B}| \).
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In the motorboat problem, if \( \phi \) is the angle made by the resultant velocity \( \mathbf{R} \) (\( 21.8\text{ km/h} \)) with the velocity vector of the motorboat (\( 25\text{ km/h} \) North), then \( \sin\phi \) is equal to:
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Match List-I with List-II: List-I List-II A Magnitude of vector \( \mathbf{A} \) in 3D space I \( \frac{\mathbf{A}}{|\mathbf{A}|} \) B Direction cosine with z-axis (\( \cos\gamma \)) II \( \sqrt{A_x^2 + A_y^2 + A_z^2} \) C Unit vector along vector \( \mathbf{A} \) III \( \frac{A_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \) D Property of zero vector \( \mathbf{0} \) IV \( 0 \cdot \mathbf{A} = \mathbf{0} \) Choose the correct answer from the options given below:
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A force vector \( \mathbf{F} \) of magnitude 50 N is resolved into two non-orthogonal component vectors making angles of \( 30^\circ \) and \( 45^\circ \) on opposite sides of \( \mathbf{F} \). Calculate the magnitude of the component vector that makes an angle of \( 30^\circ \) with \( \mathbf{F} \).
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A hiker stands on the edge of a cliff \( 490\text{ m} \) above the ground and throws a stone horizontally with an initial speed of \( 15\text{ m/s} \). Taking \( g = 9.8\text{ m/s}^2 \), the time taken by the stone to reach the ground and the speed with which it hits the ground are respectively:
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Given two vectors \( \mathbf{A} = 2\hat{\mathbf{i}} + 3\hat{\mathbf{j}} - \hat{\mathbf{k}} \) and \( \mathbf{B} = \hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}} \). Find a unit vector parallel to the linear combination vector \( \mathbf{C} = 2\mathbf{A} - 3\mathbf{B} \).
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Refer to the vector diagram shown in the figure for a person experiencing rain and wind. Rain is falling vertically with a speed \( v_r = 30\text{ m/s} \), and wind is blowing from east to west with a speed \( v_w = 10\sqrt{3}\text{ m/s} \). Calculate the magnitude of the velocity of rain relative to the ground and the angle \( \theta \) with the vertical at which the umbrella should be held to prevent getting wet.
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If the speed of a particle moving in a circular path is doubled and the radius of the circular path is halved, the centripetal acceleration becomes:
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Given below are statements regarding unit vectors: (A) A unit vector is a vector of unit magnitude and has no physical unit or dimension. (B) Unit vectors along x, y, and z axes of a rectangular coordinate system are represented by \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) respectively. (C) The magnitude of the vector sum \( \hat{i} + \hat{j} \) is equal to 2. (D) Multiplying a scalar quantity by a unit vector gives a vector quantity acting in that specific direction. Choose the correct answer from the options given below:
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Match List-I with List-II: List-I List-II A \( \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \) I Null vector B \( (\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C}) \) II Vector scaled in same direction C \( \mathbf{A} + (-\mathbf{A}) = \mathbf{0} \) III Commutative law of vector addition D \( \lambda \mathbf{A} \) (where \( \lambda > 0 \)) IV Associative law of vector addition Choose the correct answer from the options given below:
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Match List-I with List-II: List-I List-II A Magnitude of 2D vector \( \mathbf{A} = A_x \hat{i} + A_y \hat{j} \) I \( \tan^{-1}(\frac{A_y}{A_x}) \) B Unit vector \( \hat{n} \) in direction of \( \mathbf{A} \) II \( \sqrt{A_x^2 + A_y^2 + A_z^2} \) C Direction angle \( \theta \) with x-axis in 2D III \( \sqrt{A_x^2 + A_y^2} \) D Magnitude of 3D vector \( \mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) IV \( \frac{\mathbf{A}}{|\mathbf{A}|} \) Choose the correct answer from the options given below:
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Two projectiles are launched with the same initial speed \( v_o \) at complementary angles of projection \( (45^\circ + \alpha) \) and \( (45^\circ - \alpha) \). The ratio of their horizontal ranges \( R_1 / R_2 \) is:
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The maximum resultant of two vectors \( \mathbf{P} \) and \( \mathbf{Q} \) is 17 units, and their minimum resultant is 7 units. If these two vectors act at right angles (\( 90^\circ \)) to each other, what is the magnitude of their resultant?
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A particle is projected from the ground with a velocity \( u \) at an angle \( \theta \) with the horizontal. The magnitude of the change in velocity of the particle between the instant of projection and the instant it reaches maximum height is:
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For an insect moving in a circular groove of radius \( 12\text{ cm} \) with a constant angular speed of \( 0.44\text{ rad/s} \), what is the magnitude of its centripetal acceleration \( a_c \)?
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Referring to the vector triangle diagram shown for the head-to-tail addition method, if vector \( \mathbf{A} \) has a magnitude of \( 12\text{ units} \) along the horizontal and vector \( \mathbf{B} \) (perpendicular to \( \mathbf{A} \)) has a magnitude of \( 5\text{ units} \), find the magnitude of the resultant vector \( \mathbf{R} = \mathbf{A} + \mathbf{B} \).
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