Differentiation

Q.

A truck travelling due north at $20\text{m/s}$ turns west and travels at the same speed. The change in its velocity is

1. $40\text{m/s}$ northwest
2. $20\sqrt{2}\text{m/s}$ northwest
3. $40\text{m/s}$ southwest
4. $20\sqrt{2}\text{m/s}$ southwest

Q. A particle moves along a straight line such that its displacement at any time $t$ is given by $s=(t^3-6t^2+3t+4)\text{m}$. The velocity when the acceleration is zero is

1. $3\text{m/s}$
2. $-12\text{m/s}$
3. $-9\text{m/s}$
4. $42\text{m/s}$

Q. An insect wants to go from one corner of cube to diagonaly opposite corner by crawling [can't fly]. Minimum distance it has to travel to reach there if side of cube is a

1. $a+\sqrt{2}a$
2. $\sqrt{3}a$
3. $3a$
4. $\sqrt{5}a$

Q.

A train $100m$ long takes $6seconds$ to cross a man walking at $5km/h$in a direction opposite to that of the train. The speed of the train is?

Q. If $T$ is the total time of flight, $H$ is the maximum height and $R$ is the horizontal range of a projectile, then $x$ and $y$ coordinates at any time $t$ are related as (Take point of projection as origin)

1. $y=4H\left(\frac{R}{x}\right)[1-\frac{R}{x}]$
   
2. $y=4H\left(\frac{x}{R}\right)[1-\frac{x}{R}]$
   
3. $y=4H\left(\frac{T}{t}\right)[1-\frac{T}{t}]$
   
4. $y=4H\left(\frac{t}{T}\right)[1-\frac{t}{T}]$
   

Q. The equation of motion of a projectile is given by $y=12x-\frac{3}{4}{x}^2$. The horizontal component of velocity is $3\text{m/s}$. Given that $g=10{\text{m/s}}^2$, what is the range of the projectile?

1. $12.4\text{m}$
2. $21.6\text{m}$
3. $30.6\text{m}$
4. $36.0\text{m}$

Q.

Starting at time $t=0$ from the origin with speed $1m{s}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$plane so that its coordinates are related by the equation $y=\frac{x^2}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_x$and $a_y$ respectuvely. Then

1. $a_x=1m{s}^{-1}$, implies that when the particle is at the origin $a_y=1m{s}^{-1}$

2. $a_x=0$ implies $a_y=1m{s}^{-1}$ at all times

3. at $t=0$, the particles velocity points in the x-direction

4. $a_x=0$ implies that at $t=1s$, the angle between the particles velocity and the x-axis is $45°$

Q. Power supplied to a particle of mass $2\text{kg}$ varies with time as $P=\frac{3t^2}{2}$ watt. Here $t$ is in second. If velocity of particle at $t=0$ is $v=0$. The velocity of particle at time $t=2$ s will be

1. $1\text{m/s}$
2. $4\text{m/s}$
3. $2\text{m/s}$
4. $2\sqrt{2}\text{m/s}$

Q.

A car travels a distance of $1375km$in $25hrs$ . What is the speed of car in $km/hrs?$

Q. The velocity of the particle at any time t is given by $v=2t(3-t)m{s}^{-1}$. At what time is its velocity maximum?

1. $\frac{2}{3}s$
2. $3s$
3. $\frac{3}{2}s$
4. $2s$

Q. A man of height $h$ walks in a straight path towards a lamp post of height $H$ with velocity $v$. Then velocity of the edge of the shadow on the ground will be

1. $\frac{hv}{(H-h)}$
2. $\frac{Hv}{(H-h)}$
3. $\frac{hv}{(H+h)}$
4. $\frac{(H-h)}{hv}$

Q.

A man walks $8$ km in$1$hour $20$ minutes. What is the speed of the man in km/hr?

Q.

A bus is moving at a speed of $84km/hr$. How much distance will it cover in $1min45sec$?

Q.

A man cycles at $12$ km per hour. How much distance does he cover in $35$ minutes?

Q. The position vector of a moving particle in $x-y$ plane can be represented as $\overrightarrow{r}=\left(2t\hat{i}+2t^2\hat{j}\right)m$, where $t$ represents the time instant. The rate of change of $\theta$ at time $t=2s$. (where $\theta$ is the angle which its velocity vector makes with positive $x-$axis)

1. $\frac{2}{5}rad/s$
2. $\frac{2}{17}rad/s$
3. $2rad/s$
4. $4rad/s$

Q. A car travels $6\text{km}$ towards north at an angle of ${45}^{\circ }$ to the east and then travels distance of $4\text{km}$ towards north at an angle ${135}^{\circ }$ to east. How far is the point from the starting point? What angle does the straight line joining its initial and final position makes with the east?

1. $\sqrt{50}\text{km}$ and ${\tan }^{-1}\left(5\right)$
2. $10\text{km}$ and ${\tan }^{-1}\left(\sqrt{5}\right)$
3. $\sqrt{52}\text{km}$ and ${\tan }^{-1}\left(5\right)$
4. $\sqrt{52}\text{km}$ and ${\tan }^{-1}\left(\sqrt{5}\right)$

Q. An insect wants to go from one corner of cube to diagonaly opposite corner by crawling [can't fly]. Minimum distance it has to travel to reach there if side of cube is a

1. $\sqrt{3}a$
2. $a+\sqrt{2}a$
3. $3a$
4. $\sqrt{5}a$

Q. A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The $\text{displacement(S) - time (t)}$ graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale):

1. 
2. 
3. 
4. 

Q.

Rain is falling vertically with a speed of 30 m s^–1. A woman rides a bicycle with a speed of 10 m s^–1 in the north to south direction. What is the direction in which she should hold her umbrella?

Q. A particle moving along the $x-$ axis has an instantaneous speed given by $v^2=108-9x^2$. The acceleration of the particle is

1. $-9x{\text{m/s}}^2$
2. $-18x{\text{m/s}}^2$
3. $\frac{-9x}{2}{\text{m/s}}^2$
4. None of these

Q. Find the local maxima and minima of the function $y=x^3-3x+2$.

1. $6$ , $3$
2. $8$ , $4$
3. $4$ , $0$
4. $0$ , $4$

Q. A particle, initially at rest is subject to two forces; one is constant, the other is a retarding force proportional to the particle velocity. In the subsequent motion of the particle,

1. the acceleration will increase from zero to a constant value
2. the acceleration will remain constant with respect to time
3. the velocity will increase from zero to a maximum and then decrease
4. the velocity will increase from zero to a constant value

Q. A particle travels according to the equation $y=x-\frac{x^2}{2}$. Find the maximum height it achieves.

1. $\frac{1}{4}\text{m}$
2. $2\text{m}$
3. $1\text{m}$
4. $\frac{1}{2}\text{m}$

Q. The position vector of a moving particle in $x-y$ plane can be represented as $\overrightarrow{r}=\left(2t\hat{i}+2t^2\hat{j}\right)m$, where $t$ represents the time instant. The rate of change of $\theta$ at time $t=2s$. (where $\theta$ is the angle which its velocity vector makes with positive $x-$axis)

1. $\frac{2}{5}rad/s$
2. $\frac{2}{17}rad/s$
3. $2rad/s$
4. $4rad/s$

Q. The trajectory of a projectile in a vertical plane is $y=ax-b{x}^2,$ where $a$ and $b$ are constants and $x$ and $y$ are horizontal and vertical distances respectively of the projectile from the point of projection. The maximum height attained by the particle and the angle of projection from the horizontal are:

1. $\frac{b^2}{2a},{\tan }^{-1}\left(a\right)$

2. $\frac{a^2}{b},{\tan }^{-1}\left(2a\right)$

3. $\frac{a^2}{4b},{\tan }^{-1}\left(a\right)$

4. $\frac{2a^2}{b},{\tan }^{-1}\left(a\right)$

Q. The position of a particle is given by $\overrightarrow{r}=3t\hat{i}+\sqrt{3}{t}^2\hat{j}-4\hat{k}$, where $t$ is in seconds and $\overrightarrow{r}$ is in meters. Find out magnitude and direction of velocity $\overrightarrow{v}$ with horizontal at $t=\sqrt{3}\text{s}$.

1. $3\sqrt{5}\text{m/s},\theta ={\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$
2. $3\sqrt{2}\text{m/s},\theta ={\tan }^{-1}\left(3\right)$
3. $3\sqrt{5}\text{m/s},\theta ={\tan }^{-1}\left(\frac{1}{2}\right)$
4. $3\sqrt{5}\text{m/s},\theta ={\tan }^{-1}\left(2\right)$

Q. The area $A$ of a circle is related to its radius by the equation
$A=\pi r^2$. The change in area with respect to the radius is $x\times \pi$ $m$. When the radius is $5m$, find the value of $x$.

Q. Power applied to a particle varies with time as $P=[3t^2-2t+1]$ watts, where t is time in seconds. Then the change in kinetic energy of particle between time t = 2 s to t = 4 s is (in Joule) –

Q. A particle moves in $xy-$plane according to the equation $x=\alpha t$ and $y=\alpha t(1-\beta t)$ where $\alpha$ and $\beta$ are positive constants and $t$ is time.

Choose the correct option in regard to the speed of the particle.

1. The speed of the particle at time $t=\frac{1}{4\beta }$ is $\frac{\sqrt{5}}{2}\alpha$.
2. The speed of the particle at time $t=\frac{1}{4\beta }$ is $\frac{\sqrt{5}}{2}\beta$.
3. The speed of the particle at time $t=\frac{1}{4\beta }$ is $0$.
4. The speed of the particle at time $t=\frac{1}{2\beta }$ is $0$.

Q. Two cars are travelling towards each other on a straight road at velocities $15m/s$ and $16m/s$ respectively. When they are $150m$ apart, both the drivers apply the brakes and the cars decelerate at $3m/s^2$ and $4m/s^2$ until they stop. Separation between the cars when they come to rest is :

1. $86.5m$
2. $85.5m$
3. $89.5m$
4. $80.5m$