Embibe Experts Solutions for Exercise 7: Assignment

A gun fitted on the top of a moving car is aimed in the backward direction at an angle $60°$ with the horizontal. If the muzzle velocity of the bullet fired from the gun is $8 \mathrm{m} {\mathrm{s}}^{–1},$ then the speed of the car for which the bullet comes out vertically is $\mathrm{n} \mathrm{m} {\mathrm{s}}^{–1}.$ The value of $\mathrm{n}$ is _____.

A particle is moving in a circular path of radius ${\mathrm{R}}_0$ with a constant speed ${\mathrm{v}}_0.$ If the particle's speed is decreased by a factor of $\mathrm{\beta }$ i.e., $\mathrm{v}\text{'}=\frac{{\mathrm{v}}_0}{\mathrm{\beta }}$ and the radius of circular path is doubled, then, the centripetal acceleration decreases by a factor ${\mathrm{m\beta }}^{\mathrm{n}}.$ The value of $\mathrm{mn}$ is _____.

Two particles $\mathrm{A}$ and $\mathrm{B}$ are revolving on two coplanar circles with time periods $4$ second and $6$ second respectively. Time period of particle $\mathrm{A}$ with respect to $\mathrm{B}$ will be $\mathrm{n} \mathrm{s}.$ The value of $\mathrm{n}$ is _____.

Speed of a particle moving on a circular path of radius $2 \mathrm{m}$ is varying with time as $v=\left(t^2\right) \mathrm{m} {\mathrm{s}}^{–1}.$ Net acceleration of the particle at $t=2$ second is $8\sqrt{n} \mathrm{m} {\mathrm{s}}^{–2}.$ The value of $n$ is _____.

A disc of radius $1 \mathrm{m}$ is rotating about central axis with angular velocity $2 \mathrm{rad} {\mathrm{s}}^{–1}.$ If a point $\mathrm{P}$ lying on its circumference as shown is moving with velocity $4\hat{\mathrm{j}} \mathrm{m} {\mathrm{s}}^{–1},$ then speed of centre of disc is $\sqrt{\mathrm{n}} \mathrm{m} {\mathrm{s}}^{–1}.$ The value of $\mathrm{n}$ is _____.

A particle is performing motion whose position vector is given by $\overrightarrow{r}=15\left(\cos pt \hat{i}+\sin pt \hat{j}\right)\mathrm{m},$ when $p$ is in $\mathrm{rad} {\mathrm{s}}^{–1}$ and $t$ is in seconds. The magnitude of centripetal acceleration at $t=3 \mathrm{s},$ is $n{p}^2 \mathrm{m} {\mathrm{s}}^{–2}.$ The value of $n$ is _____.

A particle is performing circular motion in a circle of radius $\mathrm{R}$ with variable speed, $\mathrm{v}=\mathrm{Kt} (\mathrm{K}>0).$ Time when angle between acceleration vector and its velocity vector becomes $\frac{\mathrm{\pi }}{4},$ is $4\sqrt{\frac{\mathrm{R}}{\mathrm{nK}}}.$ The value of $\mathrm{n}$ is _____.

A particle moves along a curve $y=x^2$ with constant speed of $10 \mathrm{m} {\mathrm{s}}^{–1}.$ When the particle is at origin, then the radius of curvature of path is $n \mathrm{m}.$ The value of $n$ is _____.