Magnitude of Instantaneous Velocity

Q. Two cars P and Q starts from a point at the same time in a straight line and their positions are represented by $x_p\left(t\right)=at+b{t}^2$ and $x_Q\left(t\right)=ft-t^2$. At what time do the cars have the same velocity?

1. $\frac{f-a}{2(1+b)}$
2. $\frac{a-f}{1+b}$
3. $\frac{a+f}{2(b-1)}$
4. 
   $\frac{a+f}{2(1+b)}$

Q. The height y and the distance x along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by y = (8t - 5$t^2$)
meter and x=6t meter, where t is in second. The velocity with which the projectile is projected is

1. 6 m/sec
2. Not obtainable from the data
3. 8 m/sec
4. 10 m/sec

Q. Velocity of a particle is $\overrightarrow{v}=6\hat{i}+2\hat{j}-2\hat{k}$. The component of the velocity parallel to vector $a=\hat{i}+\hat{j}-\hat{k}$ in vector form is

1. $6\hat{i}+2\hat{j}+2\hat{k}$
2. $\hat{i}+\hat{j}+\hat{k}$
3. $2\hat{i}+2\hat{j}+2\hat{k}$
4. $6\hat{i}+2\hat{j}-2\hat{k}$

Q. Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_P\left(t\right)=\alpha t+\beta t^2$ and $X_Q\left(t\right)=ft-t^2$. At what time, both the buses have same velocity?

Q. Two projectiles are fired from the two imaginary planets A and B. The projectile is fired from the planet A with a velocity of $10m/s$ and at angle $\theta$ with horizontal. Another projectile is fired from the surface of planet B with a velocity of $6m/s$ at the same angle follows trajectory which is identical with the trajectory of projectile from planet A. What is the ratio of the acceleration due to gravity of planet A to planet B?

1. $\frac{5}{3}$
2. $\frac{3}{5}$
3. $\frac{25}{9}$
4. $\frac{9}{25}$

Q. The height y and the distance x along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by $y=(8t-5t^2)$ meter and x = 6t meter, where t is in second. The velocity with which the projectile is projected is

1. $8m/sec$
2. $6m/sec$
3. $10m/sec$
4. Not obtainable from the data.

Q.

A body is projected with an initial velocity of (5 $\hat{i}$ + 8 $\hat{j}$) m/s. The range of the projectile (g = 10 m / $s^2$)

1. 8 m

2. 30 m

3. 10 m

4. 20 m

Q. Find the velocity of a projectile at the highest point, if it is projected with a speed $15m{s}^{-2}$, in the direction ${45}^{\circ }$ above horizontal.
[take $g=10m{s}^{-2}]$

Q. A projectile is fired horizontally with a velocity of $98m/s$ from the top of a hill $490m$ high. Find the horizontal range of the projectile.

Q. The height y and the distance x along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by y = (8t - 5$t^2$)
meter and x=6t meter, where t is in second. The velocity with which the projectile is projected is

1. 8 m/sec
2. 6 m/sec
3. 10 m/sec
4. Not obtainable from the data

Q. The vertical height $y$ and horizontal distance $x$ of a projectile on a certain planet are given by $x=\left(3t\right)m,y=\left(4t\right)m$ where $t$ is in seconds. Find the speed of projection $\left(inm/s\right)$ ?

Q. A particle is moving in $x-y$ plane on a straight line its $x$ and $y$ coordinates are given as $x=({2t}^2+4)m$ and $Y=(\frac{\sqrt{3}x}{2}+8)m$ and $t$ is in second. The acceleration of the particle is

1. $\sqrt{7}{m/s}^2$
2. $\sqrt{5}{m/s}^2$
3. $2\sqrt{7}{m/s}^2$
4. None of these

Q.

The coordinates of a moving particle at any time are given by $x=a{t}^2$ and $y=b{t}^2$. The speed of the particle at any moment is

1. 

2. 

3. 

4. 

Q. The height $y$ and the distance $x$ along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by $y=(8t-5t^2)$ m and $x=6tm$, where t is in seconds. The velocity with which the projectile is projected at t = 0 is:

1. $8m{s}^{-1}$
2. $6m{s}^{-1}$
3. $10m{s}^{-1}$
4. Not obtainable from the data

Q. A body changes direction by an angle a, without changing speed (v). Change in magnitude of velocity is

1. zero
2. 2v sin a/2
3. v sin a/2
4. v sin a

Q. The height y and the distance x along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by y = (8t - 5$t^2$)
meter and x=6t meter, where t is in second. The velocity with which the projectile is projected is

1. 8 m/sec
2. 6 m/sec
3. 10 m/sec
4. Not obtainable from the data

Q. Two swimmers start from point $P$ on one bank of the river to reach $Q$ on the opposite bank. Velocity of each swimmer in still waters is $2.5km{h}^{-1}$. One of the swimmers crosses the river along the straight route $PQ$ and the other swims right angles to the stream and then walks the distance which he has been carried away by the river to get to point $Q$. Stream velocity is $2km{h}^{-1}$. If both the swimmers reach point $Q$ simultaneously, the velocity of walking of second swimmer is

1. $3km{h}^{-1}$
2. $4km{h}^{-1}$
3. $2km{h}^{-1}$
4. $3.5km{h}^{-1}$

Q. A body starts from the origin and moves along the X-axis such that the velocity at any instant is given by $(4t^3-2t)$. Where t is in second and velocity is in m/s. What is acceleration of the particle, when it is at distance 2m from the origin.

1. $28m/s^2$
2. $22m/s^2$
3. $12m/s^2$
4. $10m/s^2$

Q. A jet plane flying at a constant velocity v at a height = 8 km is tracked by a radar R located at O directly below the line of flight. If angle $\theta$ is decreasing at rate of 0.025 rad/s velocity of plane when $\theta$ = 60${}^{\circ }$ is

1. 
   1440 km/h
2. 
   960 km/h
3. 
   1920 km/h
4. 
   480 km/h

Q.

The coordinates of a moving particle at any time are given by $x=a{t}^2$ and $y=b{t}^2$. The speed of the particle at any moment is

1. 2t(a + b)

2. $2t\sqrt{(a^2-b^2)}$

3. $t\sqrt{(a^2+b^2)}$

4. $2t\sqrt{(a^2+b^2)}$

Q. A jet plane flying at a constant velocity v at a $\text{height}=8\text{km}$ is tracked by a radar R located at O directly below the line of flight. If angle $\theta$ is decreasing at rate of $0.025\text{rad/s}$, the velocity of plane when $\theta ={60}^{\circ }$ is

1. 
   1440 km/h
2. 
   960 km/h
3. 
   1920 km/h
4. 
   480 km/h

Q. A particle is moving on a circular path of radius r with uniform speed V. The magnitude of change in velocity when the particle moves from P to Q is :- $(\angle PPOQ={40}^o)$

1. $2vcos{40}^o$
2. $2vsin{40}^o$
3. $2vsin{20}^o$
4. $2vcos{20}^o$

Q.

The coordinates of a moving particle at any time are given by $x=a{t}^2$ and $y=b{t}^2$. The speed of the particle at any moment is

1. 2t(a + b)

2. $2t\sqrt{(a^2-b^2)}$

3. $t\sqrt{(a^2+b^2)}$

4. $2t\sqrt{(a^2+b^2)}$

Q. Two cars P and Q starts from a point at the same time in a straight line and their positions are represented by $x_p\left(t\right)=at+b{t}^2$ and $x_Q\left(t\right)=ft-t^2$. At what time do the cars have the same velocity?

1. $\frac{f-a}{2(1+b)}$
2. $\frac{a-f}{1+b}$
3. $\frac{a+f}{2(b-1)}$
4. 
   $\frac{a+f}{2(1+b)}$

Q. A spaceship approaches the Moon along a parabolic trajectory which is almost tangent to the Moon's surface. At the moment of the maximum approach the brake rocket was fired for a short time interval, and the spaceship was transferred into a circular orbit of a Moon satellite. Find how the spaceship velocity modulus increased in the process of braking.

Q. Velocity of a particle as a function of time is shown in graph which is a parabola. If slope of the graph at $t=0$ is $2$ units and $v=p+qt+r{t}^2$, then (Assume SI units everywhere)

1. $q=4$
2. $r=-0.5$
3. $v_0=6m/s$
4. Velocity of particle is zero at $t=2\sqrt{3}+2$ sec

Q. A body starts from the origin and moves along the $x-axis$ such that velocity at any instant is given by $(4t^3-2t)$ where $t$ is in second and velocity is in $m/s$. What is the acceleration of the particle, when it is $2m$ from the origin?

1. $28m/s^2$
2. $22m/s^2$
3. $12m/s^2$
4. $10m/s^2$

Q. Velocity of a particle as a function of time is shown in graph which is a parabola. If slope of the graph at $t=0$ is $2$ units and $v=p+qt+r{t}^2$, then (Assume SI units everywhere)

1. $q=4$
2. $r=-0.5$
3. $v_0=6m/s$
4. Velocity of particle is zero at $t=2\sqrt{3}+2$ sec

Q. A body changes direction by an angle a, without changing speed (v). Change in magnitude of velocity is

1. zero
2. 2v sin a/2
3. v sin a/2
4. v sin a

Q. The position vector of an object at any time $t$ is given by $3t^2\hat{i}+6t\hat{j}+\hat{k}$. Its velocity along y-axis has the magnitude

1. $9$
2. $6$
3. $0$
4. $6t$