Chapter: Motion in a Plane

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

If \(\begin{align}A=\cos \omega t\widehat{i}+\sin \omega t\widehat{j}\end{align}\)and \(\begin{align}B=\cos \frac{{\omega t}}{2}\widehat{i}+\sin \frac{{\omega t}}{2}\widehat{j}\end{align}\) are functions of time, then the value of \(\begin{align}t\end{align}\) at which they are orthogonal to each other, is

Six vectors a to f have the magnitudes and directions indicated in the figure. Which of the following statements is true?

\(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\) are two vectors and \(\begin{align}\theta \end{align}\) is the angle between them. If \(\begin{align}\left| {A\times \left. B \right|} \right.=\sqrt{3}\left( {A\cdot B} \right)\end{align}\) , then the value of \(\begin{align}\theta \end{align}\) is

If a vector \(\begin{align}2\widehat{i}+3\widehat{j}+8\widehat{k}\end{align}\) is perpendicular to the vector \(\begin{align}4\widehat{i}-4\widehat{i}+\alpha \widehat{k}\end{align}\) , then the value of \(\begin{align}\alpha \end{align}\) is

If \(\begin{align}\left| A \right.\times \left. B \right|=\sqrt{3}A\cdot B\end{align}\) , then the value of \(\begin{align}\left| {A+\left. B \right|} \right.\end{align}\) is

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

If a unit vector is represented by \(\begin{align}0.5\widehat{i}+0.8\widehat{j}+c\widehat{k}\end{align}\) ,then the value of c is

Which of the following is not a vector quantity?

The angle between the two vectors \(\begin{align}A=3\widehat{i}+4\widehat{j}+5\widehat{k}\end{align}\)and \(\begin{align}B=3\widehat{i}+4\widehat{j}-5\widehat{k}\end{align}\) will be

The resultant of A×0will be equal to

The angle between \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\) is \(\begin{align}\theta \end{align}\). The value of the triple product \(\begin{align}A\cdot \left( {B\times A} \right)\end{align}\) is

The magnitudes of vectors \(\begin{align}A\text{ },B\end{align}\)and \(\begin{align}C\end{align}\) are \(\begin{align}3,\text{ }4\end{align}\)and \(\begin{align}5\end{align}\) units respectively. If \(\begin{align}A+B=C\end{align}\),the angle between \(\begin{align}\text{A}\end{align}\) and \(\begin{align}\text{B}\end{align}\) is

Two bullets are fired horizontally and simultaneously towards each other from roof tops of two buildings \(\begin{align}100\,m\end{align}\) apart and of same height of \(\begin{align}200\,m\end{align}\) with the same velocity of \(\begin{align}25\,m/s\end{align}\). When and where will the two bullets collides. \(\begin{align}\left( {g=10m/{{s}^{2}}} \right)\end{align}\)

When an object is shot from the bottom of a long smooth inclined plane kept at an angle \( 60{}^\text{o}\) with horizontal, it can travel a distance \(\begin{align}{{x}_{1}}\end{align}\) along the plane. But when the inclination is decreased to \( 30{}^\text{o}\) and the same object is shot with the same velocity, it can travel \(\begin{align}{{x}_{2}}\end{align}\) distance. Then \(\begin{align}{{x}_{1}}:{{x}_{2}}\end{align}\) will be

The \(\begin{align}x\end{align}\) and \(\begin{align}y\end{align}\) coordinates of the particle at any time are \(\begin{align}x=5t-2{{t}^{2}}\end{align}\) and \(\begin{align}y=10t\end{align}\) respectively, where \(\begin{align}x\end{align}\) and \(\begin{align}y\end{align}\) are in metres and t in seconds. The acceleration of the particle at \(\begin{align}t=2s\end{align}\)is

\(\begin{align}A\end{align}\) ship \(\begin{align}A\end{align}\) is moving Westwards with a speed of \(\begin{align}10\,km{{h}^{{-1}}}\end{align}\) and a ship \(\begin{align}B\,100km\end{align}\) South of \(\begin{align}A\end{align}\), is moving Northwards with a speed of \(\begin{align}10\,km{{h}^{{-1}}}\end{align}\). The time after which the distance between them becomes shortest is

The position vector of a particle \(\begin{align}R\end{align}\) as a function of time is given by \(\begin{align}R=4\,\sin \left( {2\pi t} \right)\widehat{i}+4\cos \,\left( {2\pi t} \right)\widehat{j}\end{align}\) where \(\begin{align}R\end{align}\) is in metre, \(\begin{align}t\end{align}\) is in seconds and \(\begin{align}\widehat{i}\end{align}\) and \(\begin{align}\widehat{j}\end{align}\) denote unit vectors along \(\begin{align}x\end{align}\) and \(\begin{align}y-\end{align}\)directions, respectively. Which one of the following statements is wrong for the motion of particle?

A projectile is fired from the surface of the earth with a velocity of \(\begin{align}5\,m{{s}^{{-1}}}\end{align}\) at angle \(\begin{align}\theta \end{align}\) with the horizontal. Another projectile fired from another planet with a velocity of \(\begin{align}3\,m{{s}^{{-1}}}\end{align}\) at the same angle follows a trajectory which is identical with the trajectory of the projectile fired from the earth. The value of the acceleration due to gravity on the planet is \(\begin{align}(in\,m{{s}^{{-2}}})\end{align}\)is \(\begin{align}\left( {given,\,g=9.8\,m{{s}^{{-2}}}} \right)\end{align}\)

A particle is moving such that its position co-ordinates \(\begin{align}\left( {x,y} \right)\end{align}\) are \(\begin{align}\text{(2m,}\,\text{3m)}\end{align}\)at time \(\begin{align}\text{t=0}\end{align}\), \(\begin{align}\text{(6m, 7m)}\end{align}\)at time \(\begin{align}\text{t=2 s}\end{align}\)and \(\begin{align}\left( {13m,\,14m} \right)\end{align}\) at time \(\begin{align}t=5\,s\end{align}\). Average velocity vector \(\begin{align}\left( {{{v}_{{av}}}} \right)\end{align}\)from \(\begin{align}t=0\end{align}\)to \(\begin{align}t=5\,s\end{align}\)is

The velocity of a projectile at the initial point \(\begin{align}A\end{align}\) is \(\begin{align}(2\widehat{i}+3\widehat{j})\,m/s\end{align}\). Its velocity \(\begin{align}(in\,m/s)\end{align}\)at point \(\begin{align}B\end{align}\)is

The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is

A missile is fired for maximum range with an initial velocity of \(\begin{align}20\,m/s\end{align}\). If \(\begin{align}g=10\,m/{{s}^{2}}\end{align}\) , the range of the missile is

A particle has initial velocity \(\begin{align}(3\widehat{i}+4\widehat{j})\end{align}\) and has acceleration \(\begin{align}(0.4\widehat{i}+0.3\widehat{j})\end{align}\). Its speed after \(\begin{align}10\,s\end{align}\) is

A particle of mass \(\begin{align}\text{m}\end{align}\) is projected with velocity \(\begin{align}\text{v}\end{align}\) making an angle of \(\begin{align}45{}^\circ \end{align}\) with the horizontal. When the particle lands on the level ground, the magnitude of the change in its momentum will be

A particle starting from the origin \(\begin{align}(0,0)\end{align}\) moves in a straight line in the \(\begin{align}(x,y)\end{align}\)plane. Its co-ordinates at a later time are \(\begin{align}(\sqrt{3},3)\end{align}\)The path of the particle makes with the \(\begin{align}x-axis\end{align}\)an angle of

For angles of projection of a projectile at angles \(\begin{align}(45{}^\circ -\theta )\end{align}\) and \(\begin{align}(45{}^\circ +\theta )\end{align}\) the horizontal ranges described by the projectile are in the ratio of

Two particles are projected with same initial velocities at an angle \(\begin{align}30{}^\circ \end{align}\) and \(\begin{align}60{}^\circ \end{align}\) with the horizontal. Then,

Two particles \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\)are connected by a rigid rod \(\begin{align}AB\end{align}\). The rod slides along perpendicular rails as shown here. The velocity of \(\begin{align}A\end{align}\) to the right is \(\begin{align}10m/s\end{align}\). What is the velocity of \(\begin{align}B\end{align}\)when angle \(\begin{align}\alpha =60{}^\circ \end{align}\) ?

A bullet is fired from a gun with a speed of \(\begin{align}\text{1000}\,\,\text{m/s}\end{align}\) in order to hit a target \(\begin{align}\text{100}\,\text{m}\end{align}\) away. At what height above the target should the gun be aimed? (The resistance of air is negligible and \(\begin{align}g=10\text{ }m/{{s}^{2}}\end{align}\) )

The position vector of a particle is \(\begin{align}r=(a\,\cos \omega t)\widehat{i}+(a\,\sin \omega t)\widehat{j}\end{align}\). The velocity of the particle is

Two bodies of same mass are projected with the same velocity at an angle \(\begin{align}30{}^\circ \end{align}\) and \(\begin{align}60{}^\circ \end{align}\) respectively. The ratio of their horizontal ranges will be

The maximum range of a gun of horizontal terrain is \(\begin{align}16\,km\end{align}\). If \(\begin{align}g=10\,m{{s}^{2}}\end{align}\), then muzzle velocity of a shell must be

The speed of a swimmer in still water is \(\begin{align}20\,m/s\end{align}\). The speed of river water is \(\begin{align}10\,m/s\end{align}\) and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path the angle at which he should make his strokes w.r.t. north is given by

Two particles \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\), move with constant velocities \(\begin{align}{{v}_{1}}\end{align}\) and \(\begin{align}{{v}_{2}}\end{align}\) . At the initial moment, their position vectors are \(\begin{align}{{r}_{1}}\end{align}\) and \(\begin{align}{{r}_{2}}\end{align}\) respectively. The condition for particles \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\) for their collision is

A person swims in a river aiming to reach exactly opposite point on the bank of a river. His speed of swimming is \(\begin{align}0.5\,m/s\end{align}\)at an angle \(\begin{align}120{}^\circ \end{align}\) with the direction of flow of water. The speed of water in stream is

The speed of a boat is \(\begin{align}5\,km/h\end{align}\) in still water. It crosses a river of width \(\begin{align}1.0\,km\end{align}\) along the shortest possible path in \(\begin{align}15\,\min \end{align}\). The velocity of the river water is (in km/h)

A boat is sent across a river with a velocity of \(\begin{align}8\,km/{{h}^{{-1}}}\end{align}\). If the resultant velocity of boat is \(\begin{align}10km\,{{h}^{{-1}}}\end{align}\), then velocity of river is

A bus is moving on a straight road towards North with a uniform speed of \(\begin{align}50\text{ }km/h\end{align}\). If the speed remains unchanged after turning through \(\begin{align}90{}^\circ \end{align}\) , the increase in the velocity of bus in the turning process is

A particle moving in a circle of radius \(\begin{align}R\end{align}\) with a uniform speed takes a time \(\begin{align}T\end{align}\) to complete one revolution. If this particle were projected with the same speed at an angle \(\begin{align}\theta \end{align}\) to the horizontal, the maximum height attained by it equals \(\begin{align}4R\end{align}\). The angle of projection \(\begin{align}\theta \end{align}\) is then given by

In the given figure, \(\begin{align}a=15\,m/{{s}^{2}}\end{align}\) represents the total acceleration of a particle moving in the clockwise direction in a circle of radius \(\begin{align}R=2.5\end{align}\). m at a given instant of time. The speed of the particle is

Two stones of masses \(\begin{align}m\end{align}\) and \(\begin{align}2m\end{align}\) are whirled in horizontal circles, the heavier one in a radius \(\begin{align}\frac{r}{2}\end{align}\) and the lighter one in radius \(\begin{align}r\end{align}\). The tangential speed of lighter stone is \(\begin{align}n\end{align}\) times that of the value of heavier stone when they experience same centripetal forces. The value of \(\begin{align}n\end{align}\) is

A particle moves in a circle of radius \(\begin{align}5\,cm\end{align}\) with constant speed and time period \(\begin{align}0.2\pi s\end{align}\). The acceleration of the particle is

A car runs at a constant speed on a circular track of radius \(\begin{align}100\,m\end{align}\), taking \(\begin{align}62.8\,s\end{align}\)for every circular lap. The average velocity and average speed for each circular lap respectively is

A stone tied to the end of a string of \(\begin{align}1\,m\end{align}\) long is whirled in a horizontal circle with a constant speed. If the stone makes \(\begin{align}22\end{align}\) revolutions in \(\begin{align}44\,s\end{align}\), what is the magnitude and direction of acceleration of the stone?

The circular motion of a particle with constant speed is

A particle moves along a circle of radius \(\begin{align}\left( {\frac{{20}}{\pi }} \right)m\end{align}\) with constant tangential acceleration. If the velocity of the particle is \(\begin{align}80\,m/s\end{align}\)at the end of the second revolution after motion has begin, the tangential acceleration is

\(\begin{align}P\end{align}\) is the point of contact of a wheel and the ground. The radius of wheel is \(\begin{align}1m\end{align}\). The wheel rolls on the ground without slipping. The displacement of point \(\begin{align}P\end{align}\) when wheel completes half rotation is

A particle of mass M is revolving along a circle of radius R and another particle of mass m is revolving in a circle of radius r. If time periods of both particles are same, then the ratio of their angular velocities is

What is the linear velocity, if angular velocity vector \(\begin{align}\omega =3\widehat{i}-4\widehat{j}+\widehat{k}\end{align}\) and position vector \(\begin{align}r=5\widehat{i}-6\widehat{j}+6\widehat{k}\end{align}\)?

A body is whirled in a horizontal circle of radius \(\begin{align}20\,cm\end{align}\). It has an angular velocity of \(\begin{align}10\,rad/s\end{align}\). What is its linear velocity at any point on circular path?

When milk is churned, cream gets separated due to

An electric fan has blades of length \(\begin{align}30\,cm\end{align}\) measured from the axis of rotation. If the fan is rotating at \(\begin{align}120\,rev/\min \end{align}\), the acceleration of a point on the tip of the blade is