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A car is moving with speed of \( 150 \,\text{km/h} \) and after applying the break it will move \( 27 \,\text{m} \) before it stops. If the same car is moving with a speed of one third the reported speed then it will stop after travelling ______ \( \text{m} \) distance.

A particle is moving in a straight line such that its velocity is increasing at \( 5 \, \text{ms}^{-1} \) per meter. The acceleration of the particle is _______ \( \text{ms}^{-2} \) at a point where its velocity is \( 20 \, \text{ms}^{-1} \).

A NCC parade is going to a uniform speed of \( 9 \, \text{km}/\text{h} \) under a mango tree on which a monkey is sitting at a height of \( 19.6 \, \text{m} \). At any particular instant, the monkey drops a mango. A cadet will receive the mango whose distance form the tree at time of drop is:
(Given \( g = 9.8 \, \text{m}/\text{s}^{2} \) )

For a particle in uniform circular motion, the acceleration \( \vec{a} \) at any point \( P(R, \theta) \) on the circular path of radius \( R \) is (when \( \theta \) is measured from the positive \( x \)-axis and \( v \) is uniform speed) :

Two buses \( P \) and \( Q \) start from a point at the same time and move in a straight line and their positions are represented by \( X_P(t)=\alpha t+\beta t^2 \) and \( X_Q(t)= f t - t^2 \). At what time, both the buses have same velocity?

A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of \( 10 \, \text{m} \) in \( t \, \text{s} \), the distance travelled by the toy in the next \( t \, \text{s} \) will be:

A ball is released from a height \( h \). If \( t_1 \) and \( t_2 \) be the time required to complete first half and second half of the distance respectively. Then, choose the correct relation between \( t_1 \) and \( t_2 \).

From the top of a tower, a ball is thrown vertically upward which reaches the ground in \( 6 \, \text{s} \). A second ball thrown vertically downward from the same position with the same speed reaches the ground in \( 1.5 \, \text{s} \). A third ball released, from the rest from the same location, will reach the ground in _________ \( \text{s} \).

If \( t=\sqrt{x}+4 \), then \( \left(\dfrac{dx}{dt}\right)_{t=4} \) is:

A bullet is shot vertically downwards with an initial velocity of \( 100 \, \text{m/s} \) from a certain height. Within \( 10 \, \text{s} \), the bullet reaches the ground and instantaneously comes to rest due to the perfectly inelastic collision. The velocity-time curve for total time \( t = 20 \, \text{s} \) will be: (Take \( g = 10 \, \text{m/s}^{2} \)).

A juggler throws balls vertically upwards with same initial velocity in air. When the first ball reaches its highest position, he throws the next ball. Assuming the juggler throws \( n \) balls per second, the maximum height the balls can reach is

A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height \( h \). Find the ratio of the times in which it is at height \( \dfrac{h}{3} \) while going up and coming down respectively.

A ball is thrown vertically upwards with a velocity of \( 19.6 \, \text{m s}^{-1} \) from the top of a tower. The ball strikes the ground after \( 6 \, \text{s} \). The height from the ground up to which the ball can rise will be \( \left(\dfrac{k}{5}\right) \, \text{m} \). The value of \( k \) is _______ (use \( g = 9.8 \, \text{m/s}^2 \) ).

The velocity of the bullet becomes one third after it penetrates \( 4 \, \text{cm} \) in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at \( (4 + x) \, \text{cm} \) inside the block. The value of \( x \) is:

An object of mass \( 5 \, \text{kg} \) is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of \( 10 \, \text{N} \) throughout the motion. The ratio of time of ascent to the time of descent will be equal to: [Use \( g=10 \, \text{ms}^{-2} \)].

At time \( t = 0 \) a particle starts travelling from a height \( 7\hat{z}\,\text{cm} \) in a plane keeping \( z \) coordinate constant. At any instant of time it’s position along the \( \hat{x} \) and \( \hat{y} \) directions are defined as \( 3t \) and \( 5t^{3} \) respectively. At \( t = 1\,\text{s} \) acceleration of the particle will be:

A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is \( 60 \, \text{m} \), then the magnitude of displacement would be: (Given \( \cos 135^{\circ}=-0.7 \))

Two balls A and B are placed at the top of \( 180 \, \text{m} \) tall tower. Ball A is released from the top at \( t = 0 \, \text{s} \). Ball B is thrown vertically down with an initial velocity \( u \) at \( t = 2 \, \text{s} \). After a certain time, both balls meet \( 100 \, \text{m} \) above the ground. Find the value of \( u \) in \( \text{m s}^{-1} \). [use \( g = 10 \, \text{m s}^{-2} \) ] :

Given below are two statements. One is labeled as Assertion A and the other is labeled as Reason R.

Assertion A: two identical balls \( A \) and \( B \) thrown with same velocity \( u \) at two different angles with horizontal attained the same range \( R \). If \( A \) and \( B \) reached the maximum height \( h_1 \) and \( h_2 \) respectively, then \( R = 4\sqrt{h_1 h_2} \)

Reason R: Product of said height. \( h_1 h_2 = \left( \dfrac{u^2 \sin^2 \theta}{2g} \right)\left( \dfrac{u^2 \cos^2 \theta}{2g} \right) \)

Choose the correct answer :

Two inclined planes are placed as shown in figure. A block is projected from the point \( A \) of inclined plane \( AB \) along its surface with a velocity just sufficient to carry it to the top point \( B \) at a height \( 10 \, \text{m} \). After reaching the Point \( B \) the block slides down on inclined plane \( BC \). Time it takes to reach to the point \( C \) from point \( A \) is \( t(\sqrt{2}+1) \, \text{s} \). The value of \( t \) is _______. (use \( g = 10 \, \text{m/s}^{2} \)).

A ball is projected vertically upward with an initial velocity of \( 50 \, \text{m s}^{-1} \) at \( t = 0 \, \text{s} \). At \( t = 2 \, \text{s} \), another ball is projected vertically with same velocity. At \( t = \_\_\_\_ \, \text{s} \), second ball will meet the first ball \( (g = 10 \, \text{m s}^{-2}) \).

When a ball is dropped into a lake from a height \( 4.9 \, \text{m} \) above the water level, it hits the water with a velocity \( v \) and then sinks to the bottom with the constant velocity \( v \). It reaches the bottom of the lake \( 4.0 \, \text{s} \) after it is dropped. The approximate depth of the lake is:

A girl standing on road holds her umbrella at \( 45^\circ \) with the vertical to keep the rain away. If she starts running without umbrella with a speed of \( 15 \sqrt{2} \, \text{km h}^{-1} \), the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is:

A ball of mass \( 0.5 \, \text{kg} \) is dropped from the height of \( 10 \, \text{m} \). The height, at which the magnitude of velocity becomes equal to the magnitude of acceleration due to gravity, is __________ \( \text{m} \). [Use \( g = 10 \, \text{m} / \text{s}^{2} \)]

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JEE Main 2022 29th July Morning Shift

If \( t=\sqrt{x}+4 \), then \( \left(\dfrac{dx}{dt}\right)_{t=4} \) is: