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Train A is moving along two parallel rail tracks towards north with speed \( 72 \, \text{km/h} \) and train B is moving towards south with speed \( 108 \, \text{km/h} \). Velocity of train B with respect to A and velocity of ground with respect to B are (in \( \text{m s}^{-1} \) ) :

A particle is moving in a straight line. The variation of position ‘\( x \)’ as a function of time ‘\( t \)’ is given as \( x = (t^{3} - 6 t^{2} - 20 t + 15 )\, \text{m} \). The velocity of the body when its acceleration becomes zero is:

A train starting from rest first accelerates uniformly up to a speed of \( 80 \, \text{km/h} \) for time \( t \), then it moves with a constant speed for time \( 3t \). The average speed for the train for the duration of journey will be (in \( \text{km/h} \)):

A particle initially at rest starts moving from reference point \( x = 0 \) along x-axis, with velocity \( v \) that varies as \( v = 4\sqrt{x}\,\text{m/s} \). The acceleration of the particle is _______ \( \text{m s}^{-2} \).

A particle is moving in one dimension (along \( x \) axis) under the action of a variable force. It’s initial position was \( 16 \, \text{m} \) right of origin. The variation of its position \( (x) \) with time \( (t) \) is given as \( x = -3t^3 + 18t^2 + 16t \), where \( x \) is in \( \text{m} \) and \( t \) is in \( \text{s} \). The velocity of the particle when its acceleration becomes zero is ______ \( \text{m/s} \).

A particle moves in a straight line so that its displacement \( x \) at any time \( t \) is given by \( x^2 = 1 + t^2 \). Its acceleration at any time \( t \) is \( x^{-n} \) where \( n = \) ________.

A cyclist starts from the point \( P \) of a circular ground of radius \( 2 \, \text{km} \) and travels along its circumference to the point \( S \). The displacement of a cyclist is:

The relation between time \( 't' \) and distance \( 'x' \) is \( t = \alpha x^{2} + \beta x \), where \( \alpha \) and \( \beta \) are constants. The relation between acceleration \( (a) \) and velocity \( (v) \) is:

A light unstretchable string passing over a smooth light pulley connects two blocks of masses \( m_{1} \) and \( m_{2} \). If the acceleration of the system is \( \dfrac{g}{8} \), then the ratio of the masses \( \dfrac{m_{2}}{m_{1}} \) is:

A particle moving in a straight line covers half the distance with speed \( 6 \, \text{m/s} \). The other half is covered in two equal time intervals with speeds \( 9 \, \text{m/s} \) and \( 15 \, \text{m/s} \) respectively. The average speed of the particle during the motion is:

A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in \( t_{1} \). If it is projected vertically downwards from the same point with the same speed, it reaches the ground in \( t_{2} \). Time required to reach the ground, if it is dropped from the top of the tower, is:

Given below are two statements:
Statement (I): The mean free path of gas molecules in inversely proportional to square of molecular diameter.
Statement (II): Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.
In the light of the above statements, choose the correct answer from the options given below:

Position of an ant ($S$ in metres) moving in Y-Z plane is given by $S = 2t^2 \hat{j} + 5\hat{k}$ (where $t$ is in second). The magnitude and direction of velocity of the ant at $t = 1\text{ s}$ will be :

A body travels \( 102.5 \, \text{m} \) in \( n^{\text{th}} \) second and \( 115.0 \, \text{m} \) in \( (n + 2)^{\text{th}} \) second. The acceleration is:

The displacement and the increase in the velocity of a moving particle in the time interval of \( t \) to \( (t+1) \, \text{s} \) are \( 125 \, \text{m} \) and \( 50 \, \text{m/s} \), respectively. The distance travelled by the particle in \( (t+2)^{\text{th}} \, \text{s} \) is________ \( \text{m} \).

A body starts moving from rest with constant acceleration covers displacement \( S_{1} \) in first \( (p-1) \) seconds and \( S_{2} \) in first \( p \) seconds. The displacement \( S_{1} + S_{2} \) will be made in time:

A ball rolls off the top of a stairway with horizontal velocity \( u \). The steps are \( 0.1 \, \text{m} \) high and \( 0.1 \, \text{m} \) wide. The minimum velocity \( u \) with which that ball just hits the step \( 5 \) of the stairway will be \( \sqrt{x} \, \text{m s}^{-1} \) where \( x = \) ________________ [use \( g = 10 \, \text{m/ s}^{2} \) ].

If a rubber ball falls from a height \( h \) and rebounds upto the height of \( \frac{h}{2} \). The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are:

A body moves on a frictionless plane starting from rest. If \( S_n \) is distance moved between \( t = n - 1 \) and \( t = n \) and \( S_{n-1} \) is distance moved between \( t = n - 2 \) and \( t = n - 1 \), the ratio \( \frac{S_{n-1}}{S_n} \) is \( \left(1 - \frac{2}{x}\right) \) for \( n = 10 \). The value of \( x \) is __________.

A body starts falling freely from height \( H \) hits an inclined plane in its path at height \( h \). As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of \( \frac{H}{h} \) for which the body will take the maximum time to reach the ground is \( \_\_\_\_\_\_\_\_ \).

A clock has \( 75 \, \text{cm} \), \( 60 \, \text{cm} \) long second hand and minute hand respectively. In \( 30 \) minutes duration the tip of second hand will travel \( x \) distance more than the tip of minute hand. The value of \( x \) in meter is nearly. [Take : \( \pi = 3.14 \)]

A body falling under gravity covers two points A and B separated by \( 80 \, \text{m} \) in \( 2 \, \text{s} \). The distance of upper point A from the starting point is __________ m (use \( g = 10 \, \text{m s}^{-2} \)).

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JEE Main 2024 6th April Evening Shift

A particle moves in a straight line so that its displacement \( x \) at any time \( t \) is given by \( x^2 = 1 + t^2 \). Its acceleration at any time \( t \) is \( x^{-n} \) where \( n = \) ________.