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Physics · MOTION IN A PLANE
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Physics · MOTION IN A PLANE
A person moves with a velocity of \( 3\text{ m/s} \) due East, while wind is blowing with a velocity of \( 4\text{ m/s} \) due North. The magnitude of the velocity of the wind relative to the person is:
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The position vector of a particle moving in the \( x\text{-}y \) plane is given by \( \mathbf{r}(t) = (A\cos\omega t)\hat{i} + (A\sin\omega t)\hat{j} \), where \( A \) and \( \omega \) are positive constants. What is the angle between its velocity vector \( \mathbf{v}(t) \) and acceleration vector \( \mathbf{a}(t) \)?
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Two projectiles are thrown with the same initial speed \( v_0 \) from ground level at complementary angles of projection \( \theta_1 = 45^\circ + \alpha \) and \( \theta_2 = 45^\circ - \alpha \) (where \( 0 < \alpha < 45^\circ \)). If \( h_1 \) and \( h_2 \) are the maximum heights attained by them respectively, what is the ratio \( \frac{h_1}{h_2} \)?
Read answer →Physics · MOTION IN A PLANE
Two vectors of equal magnitude \( F \) have a resultant of magnitude \( F \). The angle \( \theta \) between the two vectors is:
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A cricket ball is thrown at a speed of \( 28\text{ m/s} \) in a direction \( 30^\circ \) above the horizontal. Taking \( g = 9.8\text{ m/s}^2 \), the maximum height \( h_m \), total time of flight \( T_f \), and horizontal distance from thrower to landing point \( R \) are respectively:
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A vector \( \mathbf{A} = 2\hat{\mathbf{i}} + 2\sqrt{3}\hat{\mathbf{j}} \) is rotated counter-clockwise about the origin by an angle of \( 30^\circ \) in the x-y plane to obtain a new vector \( \mathbf{B} \). Express vector \( \mathbf{B} \) in component form.
Read answer →Physics · MOTION IN A PLANE
A particle's position vector as a function of time \( t \) is given by \( \mathbf{r}(t) = (3t^2 - 2t)\hat{i} + (4t - 5)\hat{j} \) in SI units. What is the magnitude of the average velocity vector between \( t = 1\text{ s} \) and \( t = 3\text{ s} \)?
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A particle is projected from ground level with speed \( u \) at an angle \( \theta \) with the horizontal. At what fraction of its maximum height \( h_m \) is the vertical component of its velocity equal to half of its initial vertical component?
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Given two vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} - \hat{k} \) and \( \mathbf{B} = \hat{i} - \hat{j} + 5\hat{k} \), the magnitude of their resultant vector \( \mathbf{R} = \mathbf{A} + \mathbf{B} \) is:
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The trajectory equation of a projectile launched from origin is \( y = \sqrt{3}x - \frac{1}{20}x^2 \) where x and y are in meters. Taking \( g = 10\text{ m/s}^2 \), the initial velocity \( v_o \) of projection is:
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Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) have magnitudes of \( 9\text{ units} \) and \( 12\text{ units} \) respectively and act at right angles to each other. The magnitude of their vector sum \( \mathbf{A} + \mathbf{B} \) is:
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For the particle described in the previous question, what is its speed at time \( t = 6\text{ s} \)?
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Two forces of equal magnitude \( F \) act on a particle such that the angle between them is \( 120^\circ \). The magnitude of their resultant force is:
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A particle is projected from ground level with speed \( v_0 \) at an angle \( \theta \) above horizontal (\( \tan\theta > 1 \)). The time after projection when the direction of motion of the projectile makes an angle of \( 45^\circ \) with the horizontal is:
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A force vector is specified as \( \mathbf{F} = (6\hat{\mathbf{i}} + 2\hat{\mathbf{j}} + 3\hat{\mathbf{k}})\text{ N} \). What is the cosine of the angle that this force vector makes with the positive y-axis?
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Given below are two statements: Statement I: Vector addition obeys the commutative law, such that \( \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \). Statement II: Vector subtraction also obeys the commutative law, such that \( \mathbf{A} - \mathbf{B} = \mathbf{B} - \mathbf{A} \). In the light of the above statements, choose the most appropriate answer from the options given below:
Read answer →Physics · MOTION IN A PLANE
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): The magnitude of the displacement vector of an object moving between two points is either less than or equal to the actual path length traversed by the object. Reason (R): Displacement is the shortest straight-line distance joining the initial and final positions, whereas path length is the total distance traveled along the actual trajectory. In the light of the above statements, choose the correct answer from the options given below:
Read answer →Physics · MOTION IN A PLANE
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): In uniform circular motion, the acceleration vector of the body is not constant even though its magnitude remains constant. Reason (R): The direction of centripetal acceleration continuously changes as it always points towards the center of the circular path. In the light of the above statements, choose the correct answer from the options given below:
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Match List-I with List-II: List-I List-II A Time of flight \( T_f \) I \( \frac{2v_o\sin\theta_o}{g} \) B Maximum height \( h_m \) II \( \frac{v_o^2\sin^2\theta_o}{2g} \) C Horizontal range \( R \) III \( \frac{v_o^2\sin 2\theta_o}{g} \) D Angle of projection for maximum range IV \( 45^\circ \) Choose the correct answer from the options given below:
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Given below are statements regarding projectile motion: (A) The path of a projectile is parabolic. (B) At the highest point of projectile motion, the velocity vector and acceleration vector are mutually perpendicular. (C) The time of flight depends only on the horizontal component of initial velocity. (D) The horizontal range is maximum when the projection angle is \( 45^\circ \). Which of the above statements are correct?
Read answer →Physics · MOTION IN A PLANE
Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) of equal magnitude \( A \) are inclined at an angle \( \theta \). If the magnitude of their vector sum \( |\mathbf{A} + \mathbf{B}| \) is \( \sqrt{3} \) times the magnitude of their vector difference \( |\mathbf{A} - \mathbf{B}| \), determine the angle \( \theta \).
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Given two vectors \( \mathbf{A} = 5\hat{i} + 8\hat{j} \) and \( \mathbf{B} = 2\hat{i} + 4\hat{j} \), the magnitude of the vector difference \( \mathbf{R} = \mathbf{A} - \mathbf{B} \) is equal to:
Read answer →Physics · MOTION IN A PLANE
