A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2sin\pi t\)
where \(x,~y,~z\)  are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:

1. \(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \) ms-2
2. \(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\pi \hat{\mathrm{k}} \) ms-2
3. \(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} \) ms-2
4. \(12 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} \) ms-2

Subtopic:  Acceleration |
 67%
From NCERT
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The coordinates of a moving particle at a time t, are given by, x = 5sin10t, y = 5cos10t. The speed of the particle is:

(1)  25

(2)  50

(3)  10

(4)  502

Subtopic:  Speed & Velocity |
From NCERT
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A particle moves so that its position vector is given by, \(\vec{r}=cos \omega t ~\hat{x}+ sin \omega t~ \hat{y},\) where \(\omega\) is a constant. Which of the following is true?
 
1. velocity and acceleration both are parallel to \(\vec{r}.\)
2. velocity is perpendicular to \(\vec{r}\) and acceleration is directed towards to origin.
3. velocity is parallel to \(\vec{r}\) and acceleration is directed away from the origin.
4. velocity and acceleration both are perpendicular to \(\vec{r}.\)
Subtopic:  Circular Motion |
 77%
From NCERT
NEET - 2016
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Two particles \(\mathrm{A}\) and \(\mathrm{B}\), move with constant velocities \(\overrightarrow{{v}_1}\) and \(\overrightarrow{{v}_2}\) respectively. At the initial moment, their position vectors are \(\overrightarrow{{r}_1}\) and \(\overrightarrow{{r}_2}\) respectively. The condition for particles \(\mathrm{A}\) and \(\mathrm{B}\) for their collision will be:
1.\(\frac{\vec{r_1}-\vec{r_2}}{\left|\vec{r_1}-\vec{r_2}\right|}=\frac{\vec{v_2}-\vec{v_1}}{\left|\vec{v_2}-\vec{v_1}\right|}\)   
2. \(\vec{r_1} \cdot \vec{v_1}=\vec{r_2} \cdot \vec{v_2}\)   
3. \(\vec{r_1} \times \vec{v_1}=\vec{r_2} \times \vec{v_2}\)     
4. \(\vec{r_1}-\vec{r_2}=\vec{v_1}-\vec{v_2}\) 

Subtopic:  Relative Motion |
 69%
From NCERT
NEET - 2015
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The position vector of a particle \(\vec{R}\) as a function of time \(t\) is given by;
\(\overrightarrow{\mathrm{R}}=4 \sin (2 \pi \mathrm{t}) \hat{\mathrm{i}}+4 \cos (2 \pi \mathrm{t}) \hat{\mathrm{j}}\)
Where \(R\) is in meters, \(t\) is in seconds and \(\mathrm{\hat{i},\hat{j}}\) denotes unit vectors along \(\mathrm{x}\) and \(\mathrm{y}\)-directions, respectively. Which one of the following statements is wrong for the motion of the particle?

1. acceleration is along \(-\overrightarrow{R}\).
2. magnitude of the acceleration vector is \(\frac{v^2}{R}\), where \(v\) is the velocity of the particle.
3. magnitude of the velocity of the particle is \(8\) m/s.
4. path of the particle is a circle of radius \(4\) m.
Subtopic:  Circular Motion |
 60%
From NCERT
NEET - 2015
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The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is:
              

1. \(-2\hat i+3\hat j~\) 2. \(2\hat i-3\hat j~\)
3. \(2\hat i+3\hat j~\) 4. \(-2\hat i-3\hat j~\)
Subtopic:  Projectile Motion |
 84%
From NCERT
AIPMT - 2013
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A particle moves in a circle of radius \(5\) cm with constant speed and time period \(0.2\pi\) s. The acceleration of the particle is:

1. \(25\) m/s2 2. \(36\) m/s2
3. \(5\) m/s2 4. \(15\) m/s2
Subtopic:  Circular Motion |
 80%
From NCERT
AIPMT - 2011
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A missile is fired for a maximum range with an initial velocity of 20 m/s. If g= 10 m/s2, then the range of the missile will be:
1. 50 m
2. 60 m
3. 20 m
4. 40 m

Subtopic:  Projectile Motion |
 83%
From NCERT
AIPMT - 2011
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A particle has initial velocity \((3\hat{i}+4\hat{j})\) and has acceleration \((0.4\hat{i}+0.3\hat{j}).\) Its speed after \(10\) s is:
1. \(7\) units
2. \(7\sqrt{2}\) units
3. \(8.5\) units
4. \(10\) units
Subtopic:  Uniformly Accelerated Motion |
 85%
From NCERT
AIPMT - 2010
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A particle of mass m is projected with velocity v making an angle of 45° with the horizontal. When the particle lands on level ground, the magnitude of change in its momentum will be:

1.  2mv

2.  mv/2

3.  mv2

4.  zero

Subtopic:  Projectile Motion |
 56%
From NCERT
AIPMT - 2008
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