Motion in a Straight Line

Consider a particle moving along a straight line, whose position as a function of time is given by $s(t)=\alpha t^2-\beta t +\gamma$, where $\alpha=1 \mathrm{~ms}^{-2}, \beta=6 \mathrm{~ms}^{-1}$ and $\gamma=5 \mathrm{~m}$. The average spee...

The following plots show variation of velocity $(v)$ with time $(t)$ of a ball thrown vertically upward, and falling back. Which of the following plots is/are correct?

When a ruler falls vertically, 5 different persons catch it with different reaction times. ( $g=9.8 \mathrm{~m} \mathrm{~s}^{-2}$ )
A. Person A has reaction time of 0.20 s .
B. Person $B$ has reaction time of 0.22 s .
C. Person C has reacti...

Two cities $X$ and $Y$ are connected by a regular bus service with a bus leaving in either direction every $T$ min. A girl is driving scooty with a speed of $60 \mathrm{~km} / \mathrm{h}$ in the direction $X$ to $Y$ notices that a bus goes ...

In some appropriate units, time $(t)$ and position $(x)$ relation of a moving particle is given by $t=x^2+x$. The acceleration of the particle is

A particle is moving along \(x\)-axis with its position (x) varying with time \((t)\) as \(x=\alpha t^4+\beta t^2+\gamma t+\delta\). The ratio of its initial velocity to its initial acceleration, respectively, is:

The velocity \((v)-\) time \((t)\) plot of the motion of a body is shown below:

The acceleration \((a)-\) time \((t)\) graph that best suits this motion is :

The position of a particle is given by
\(\vec{r}(t)=4 t \hat{i}+2 t^2 \hat{j}+5 \hat{k}\)
where \(\mathrm{t}\) is in seconds and \(\mathrm{r}\) in metre. Find the magnitude and direction of velocity \(v(t)\), at \(t=1 \mathrm{~s}\), with r...

A vehicle travels half the distance with speed \(v\) and the remaining distance with speed \(2 v\). Its average speed is :

The position-time (x - t) graph for positive acceleration is

The displacement-time graphs of two moving particles make angles of 30\(^\circ\) and 45\(^\circ\) with the x-axis as shown in the figure. The ratio of their respective velocity is

The ratio of the distances travelled by a freely falling body in the 1st, 2nd, 3rd and 4th second

A small block slides down on a smooth inclined plane, starting from rest at time t = 0. Let Sn be the distance travelled by the block in the interval t = n \(-\) 1 to t = n. The the ratio \({{{S_n}} \over {{S_{n + 1}}}}\) is :

A ball is thrown vertically downward with a velocity of 20 m/s from the top of a tower. It hits the ground after some time with a velocity of 80 m/s. The height of the tower is : (g = 10 m/s2)

The speed of a swimmer in still water is 20 m/s. The speed of river water is 10 m/s 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 st...

Preeti reached the metro station and found that the escalator was not working. She walked up the sationary escalator in time t1. On another days, if she remains stationary on the the moving escalator, then the escalator takes her up in t...

Two cars P and Q start from a point at the same time in a straight line and their positions are represented by
xP(t) = (at + bt2) and xQ(t) = (ft \(-\) t2).
At what time do the cars have the same velocity ?

If the velocity of a particle is v = At + Bt2, where A and B are constants, then the distance travelled by it between 1 s and 2 s is

A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to \(v\left( x \right) = \beta {x^{ - 2n}}\), where \(\beta\) and n are constants and x is the position of the particle. The acceleration of t...

The displacement 'x' (in meter) of a particle of mass 'm' (in kg) moving in one dimension under the action of a force, is related to time 't' (in sec) by t = \(\sqrt x + 3\). The displacement of the particle when its velocity is zero, wil...

A stone falls freely under gravity. It covers distances h1, h2 and h3 in the first 5 seconds, the next 5 seconds and the next 5 seconds respectively. The relation between h1, h2 and h3 is

The motion of a particle along a straight line is described by equation x = 8 + 12t \(-\) t3 where x is in metre and t in second. The retardation of the particle when its velocity becomes zero is

A boy standing at the top of a tower of 20 m height drops a stone. Assuming g = 10 m s\(-\)2, the velocity with which it hits the ground is

A particle covers half of its total distance with speed v1 and the rest half distance with speed v2. Its average speed during the complete journey is

A particle moves a distance x in time t according to equation x = (t + 5)\(-\)1. The acceleration of particle is proportional to

A ball is dropped from a high rise platform at t = 0 starting from rest. After 6 seconds another ball is thrown downwards from the same platform with a speed v. The two balls meet at t = 18 s. What is the value of v?
(Take g = 10 m/s2)

A particle starts its motion from rest under the action of a constant force. If the distance covered in first 10 seconds is S1 and that covered in the first 20 seconds is S2, then

A bus is moving with a speed of 10 ms\(-\)1 on a straight road. A scooterist wishes to overtake the bus in 100 s. If the bus is at a distance of 1 km from the scooterist, with what speed should the scooterist chase the bus ?

The distance travelled by a particle starting from rest and moving with an acceleration \({4 \over 3}\)m s\(-\)2, in the third second is

A particle moves in a straight line with a constant acceleration. It changes its velocity from 10 ms\(-\)1 to 20 ms\(-\)1 while passing through a distance 135 m in t second. The value of t is

A car moves from X to Y with a uniform speed vu and returns to Y with a uniform speed vd. The average speed for this round trip is

A particle moving along x-axis has acceleration f, at time t, given by f = f0\(\left( {1 - {t \over T}} \right),\) where f0 and T are constants.The particle at t = 0 has zero velocity. In the time interval between t = 0 and the instant when...

The positions x of a particle with respect to time t along x-axis is given by x = 9t2 \(-\) t3 where x is in metres and t in seconds. What will be the position of this particle when it achieves maximum speed along the + x direction ?

A car runs at a constant speed on a circular track of radius 100 m, taking 62.8 seconds for every circular lap. The average velocity and average speed for each circular lap respectively is

A particle moves along a straight line OX. At a time t (in seconds) the distance x (in metres) of the particle from O is given by x = 40 + 12t \(-\) t3. How long would the particle travel before coming to rest ?

Two bodies A (of mass 1 kg) and B (of mass 3 kg) are dropped from heights of 16 m and 25 m, respectively. The ratio of the time taken by them to reach the ground is

A ball is thrown vertically upward. It has a speed of 10 m/sec when it has reached one half of its maximum height. How high does the ball rise?
(Take g = 10 m/s2.)

The displacement x of a particle varies with time t as x = ae\(-\)at + be\(\beta\)t, where a, b, \(\alpha\) and \(\beta\) are positive constants. The velocity of the particle will

A man throws balls with the same speed vertically upwards one after the other at an interval of 2 seconds. What should be the speed of the throw so that more than two balls are in the sky at any time ? (Given g = 9.8 m/s2)

If a ball is thrown vertically upwards with speed u, the distance covered during the last t seconds of its ascent is

A particle is thrown vertically upward. Its velocity at half of the height is 10 m/s, then the maximum height attained by it (g = 10 m/s2)

Motion of a particle is given by equation s = (3t3 + 7t2 + 14t + 8) m. The value of acceleration of the particle at t = 1 sec is