Instantaneous acceleration

Q. A boy stretches a stone against the rubber tape of a catapult or 'gulel' (a device used to detach mangoes from the tree by boys in Indian villages) through a distance of $24\text{cm}$ before leaving it. The tape returns to its initial normal position, accelerating the stone over the stretched length. The stone leaves the gulel with a speed of $2.2\text{m/s}$. Assuming that acceleration is constant while the stone was being pushed by the tape, find the magnitude.

1. $8.1{\text{m/s}}^2$
2. $9.1{\text{m/s}}^2$
3. $10.1{\text{m/s}}^2$
4. $11.1{\text{m/s}}^2$

Q. In the one-dimensional motion of a particle, the relation between position $x$ and time $t$ is given by $x^2+2x=t$ , where $x>0$. Choose the correct statement.

1. The retardation of the particle is $\frac{1}{4(x+1{)}^3}$.
2. The velocity of the particle is $\frac{1}{(x+1{)}^3}$.
3. Both are correct
4. Both are wrong

Q. The displacement of a particle varies with time t, $x=a{e}^{-at}+b{e}^{\beta t}$ where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will

1. Go on decreasing with time
2. Be independent of $\alpha$ and $\beta .$
3. Drop to zero when$\alpha =\beta$
4. Go on increasing with time

Q. A particle moves along x-axis and its acceleration at any time $t$ is $a=2sin\left(\pi t\right)$, where $t$ is in seconds and $a$ is in $m/s^2$. The initial velocity of particle (at time $t=0$) is $u=0$. The distance travelled (in meters) by the particle from time $t=0$ to $t=1\text{s}$ will be

1. $\frac{2}{\pi }$
2. $\frac{1}{\pi }$
3. $\frac{4}{\pi }$
4. None of these

Q. The relation between time $t$ and distance $x$ is $t=\alpha x^2+\beta x$ where $\alpha$ and $\beta$ are constants. The retardation is

1. $2\alpha v^3$
2. $2\beta v^3$
3. $2\alpha \beta v^3$
4. $2{\beta }^2{v}^3$

Q. A particle is moving along a straight-line path according to the relation
$S^2=a{t}^2+2bt+c$
$S$ represents the displacement covered in t seconds and $a,b,c$ are constants. The acceleration of the particle varies as

1. $S^{-3}$
2. $S^{3/2}$
3. $S^{-2/3}$
4. $S^2$

Q. A particle moves along a straight line. Its position at any instant is given by $x=32t-\frac{8t^3}{3}$ where $x$ is in $\text{m}$ and $t$ in $\text{s}$. The acceleration of the particle at the instant when particle is at rest will be

1. $-16{\text{m/s}}^2$
2. $-32{\text{m/s}}^2$
3. $16{\text{m/s}}^2$
4. $32{\text{m/s}}^2$

Q. The motion of a body is given by the equation $\frac{dv\left(t\right)}{dt}=6.0-3v\left(t\right).$ Where v(t) is speed in $\frac{m}{s}$ and t in sec. If body was at rest at t = 0

1. The terminal speed is $3.0\frac{m}{s}$
2. The speed varies with the time as $v\left(t\right)=2(1-e^{-3t})\frac{m}{s}$
3. The speed is $0.1\frac{m}{s}$ when the acceleration is half the initial value
4. The magnitude of the initial acceleration is $0.5\frac{m}{s^2}$

Q. A boy stretches a stone against the rubber tape of a catapult or 'gulel' (a device used to detach mangoes from the tree by boys in Indian villages) through a distance of $24\text{cm}$ before leaving it. The tape returns to its initial normal position, accelerating the stone over the stretched length. The stone leaves the gulel with a speed of $2.2\text{m/s}$. Assuming that acceleration is constant while the stone was being pushed by the tape, find the magnitude.

1. $8.1{\text{m/s}}^2$
2. $9.1{\text{m/s}}^2$
3. $10.1{\text{m/s}}^2$
4. $11.1{\text{m/s}}^2$

Q. When acceleration be function of velocity as $a=f\left(v\right)$, then

1. The displacement $\left(x\right)=\int \frac{vdv}{f\left(v\right)}$
2. The acceleration may be constant
3. The slope of acceleration versus velocity graph may be constant.
4. (a) and (c) are correct.

Q. The acceleration of a particle is increasing linearly with time t as bt . The particle starts from the origin with an initial velocity $v_o$. The distance travelled by the particle in time t will be

1. $v_{0}t+\frac{1}{3}b{t}^2$
2. $v_{0}t+\frac{1}{3}b{t}^3$
3. $v_{0}t+\frac{1}{6}b{t}^3$
4. $v_{0}t+\frac{1}{2}b{t}^2$

Q. Given below are the equations of motion of four particles $A,B,C$ and $D$.
$x_A=6t-3$;
$x_B=4t^2-2t+3$;
$x_C=3t^3-2t^2+t-7$;
$x_D=7\cos 60°-3\sin 30°$
Which of these four particles move with uniform non-zero acceleration?

1. $A$
2. $B$
3. $C$
4. $D$

Q. A particle is moving along positive $x$-axis and at $t=0$, the particle is at $x=0$. The acceleration of the particle is a function of time. The acceleration at any time $t$ is given by $a=2(1–[t\left]\right)$ where $\left[t\right]$ is the greatest integer function . Assuming that the particle is at rest initially, the average speed of the particle for the interval $t=0s\text{to}t=4s$ is

1. $1m/s$
2. $0.5m/s$
3. $2m/s$
4. $1.5m/s$

Q. A particle moves along a straight line. Its position at any instant is given by $x=32t-\frac{8t^3}{3}$ where $x$ is in $\text{m}$ and $t$ in $\text{s}$. The acceleration of the particle at the instant when particle is at rest will be

1. $-16{\text{m/s}}^2$
2. $-32{\text{m/s}}^2$
3. $16{\text{m/s}}^2$
4. $32{\text{m/s}}^2$

Q. The displacement $x$ of a particle depends on time $t$ as $x=\alpha t^2-\beta t^3.$ Then,

1. The particle will return to its starting point after time $\frac{\alpha }{\beta }$
2. The particle will come to rest after time $\frac{2\alpha }{3\beta }$
3. The initial velocity of the particle was zero but its initial acceleration was not zero
4. No net force will act on the particle at $t=\frac{\alpha }{3\beta }$

Q. Velocity of the particle in a rectillinear motion is given as $v=2t^2-5t$ in $\text{m/s}$. Find
(i) Acceleration at $4$ sec.
(ii) Average accleration in $4$ sec.

1. (i) $11m/s^2$
   (ii) $3m/s^2$
2. (i) $7m/s^2$
   (ii) $2m/s^2$
3. (i) $17m/s^2$
   (ii) $6m/s^2$
4. (i) $16m/s^2$
   (ii) $7m/s^2$

Q. A particle is moving along x-axis whose instantaneous speed is $v=\sqrt{108-9x^2}$. 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