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Physics · MOTION IN A PLANE
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Physics · MOTION IN A PLANE
A particle has an initial velocity \( \mathbf{v}_0 = (4\hat{i} + 3\hat{j})\text{ m/s} \) and a constant acceleration \( \mathbf{a} = (-2\hat{i} + \hat{j})\text{ m/s}^2 \). At what time \( t \) will the instantaneous velocity vector \( \mathbf{v}(t) \) be perpendicular to its initial velocity vector \( \mathbf{v}_0 \)?
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A particle is projected from the ground with an initial velocity \( \mathbf{v}_0 = u\hat{i} + v\hat{j} \). If the horizontal range of the projectile is equal to twice its maximum height (\( R = 2H \)), then the ratio \( \frac{v}{u} \) is:
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Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) have equal magnitudes \( A \) and are separated by an angle \( \theta \). If their resultant vector \( \mathbf{R} \) makes an angle \( \alpha \) with vector \( \mathbf{A} \), determine the relation between \( \alpha \) and \( \theta \), and the expression for the magnitude of \( \mathbf{R} \).
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A particle moves in a circular path of radius \( R = 2\text{ m} \) with a constant speed of \( v = 10\text{ m/s} \). The magnitude of centripetal acceleration of the particle is:
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Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) have magnitudes of \( 5\text{ units} \) and \( 12\text{ units} \) respectively. If the magnitude of their resultant \( \mathbf{R} = \mathbf{A} + \mathbf{B} \) is \( 13\text{ units} \), the angle \( \theta \) between \( \mathbf{A} \) and \( \mathbf{B} \) is:
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A particle starts from the origin at \( t = 0 \) with an initial velocity \( \mathbf{v}_0 = 5.0\hat{i}\text{ m/s} \) and moves in the \( x\text{-}y \) plane under a constant acceleration \( \mathbf{a} = (3.0\hat{i} + 2.0\hat{j})\text{ m/s}^2 \). When the \( x \)-coordinate of the particle is \( 84\text{ m} \), its \( y \)-coordinate and its speed are respectively:
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A vector \( \mathbf{A} \) in three dimensions makes angles \( \alpha = 60^\circ \) and \( \beta = 60^\circ \) with the x and y axes respectively. What angle \( \gamma \) does it make with the z-axis?
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If a vector \( \mathbf{A} = 4\hat{i} - 3\hat{j} \) is multiplied by a scalar \( \lambda = 0 \), what is the magnitude of the resulting vector?
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The position vector of a particle moving in the xy-plane is given by \( \vec{r}(t) = (4t^2 - 2t)\hat{i} + (3t^3 + 5)\hat{j} \) in SI units. The magnitude of the displacement vector of the particle between time \( t = 1\text{ s} \) and \( t = 2\text{ s} \) is:
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In the velocity diagram shown for a rain-wind situation, rain falls vertically with speed \( v_r = 35\text{ m/s} \) and wind blows horizontally from east to west with speed \( v_w = 12\text{ m/s} \). What is the exact value of \( \tan\theta \), where \( \theta \) is the angle made with the vertical direction at which an umbrella must be held?
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A man standing on a road observes rain falling vertically at a speed of \( 20\text{ m/s} \). When he starts running horizontally at \( 20\text{ m/s} \), the rain appears to hit him at an angle \( \theta_1 \) with the vertical. If he doubles his running speed to \( 40\text{ m/s} \), the rain appears to hit him at an angle \( \theta_2 \) with the vertical. Calculate the ratio \( \frac{\tan\theta_2}{\tan\theta_1} \).
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Two forces of magnitudes 8 N and 15 N act at a point. If their resultant magnitude is 17 N, calculate the angle \( \theta \) between the two force vectors and the angle \( \beta \) that the resultant makes with the 8 N force.
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Which of the following represents the unit vector \( \hat{n} \) in the direction of vector \( \mathbf{A} = 6\hat{i} + 8\hat{j} \)?
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A particle moves in the xy-plane such that its position coordinates as functions of time are \( x(t) = 5\sin(2t) \) and \( y(t) = 5\cos(2t) \) in meters. The magnitude of the acceleration of the particle is:
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For the insect moving in the circular groove of radius \( 12\text{ cm} \) with angular speed \( \omega = 0.44\text{ rad/s} \), the magnitude of its acceleration is approximately:
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A hiker stands at the edge of a cliff \( 490\text{ m} \) high 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 is:
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A projectile is launched from ground level with an initial speed \( v_o = 20\text{ m/s} \) at an angle \( \theta_o = 30^\circ \) above the horizontal. Taking \( g = 10\text{ m/s}^2 \), the total time of flight is:
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For a projectile fired from the ground with initial speed \( v_0 \) at an angle \( \theta_0 \) above the horizontal, the ratio of its kinetic energy at the highest point to its initial kinetic energy is \( 1/4 \). What is the time of flight of this projectile in terms of \( v_0 \) and \( g \)?
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The magnitude of a three-dimensional position vector \( \mathbf{r} = 3\hat{i} + 4\hat{j} + 12\hat{k} \) is:
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Given below are two statements: Statement I: In projectile motion, the horizontal component of velocity remains constant throughout the motion in the absence of air resistance. Statement II: In uniform circular motion, the centripetal acceleration vector remains constant in both magnitude and direction. In the light of the above statements, choose the most appropriate answer from the options given below:
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A position vector \( \mathbf{r} \) in the x-y plane has a magnitude of 10 m and makes an angle of \( 30^\circ \) with the positive y-axis towards the positive x-axis. What are its rectangular components \( r_x \) and \( r_y \)?
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The maximum height reached by a projectile launched with speed \( v_o \) at an angle \( 60^\circ \) to the horizontal is \( H_1 \). If the same projectile is launched with the same speed \( v_o \) at an angle \( 30^\circ \), its maximum height is \( H_2 \). The ratio \( H_1 / H_2 \) is:
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