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Q 4. 22  $\hat i$ and $\hat j$ are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors $\hat i + \hat j$, and $\hat i - \hat j$? What are the components of a vector$A=2\hat i +3 \hat j$ along the directions of $\hat i + \hat j$ and $\hat i - \hat j$? [You may use graphical method]

Let A be a vector such that:-                  Then the magnitude of vector A is given by   :             Now let us assume that the angle made between vector A and x-axis is . Then we have:-                                                                                   Similarly, let B be a vector such that:-        The magnitude of vector B is     :                        Let  be the angle...

Q. 4.32 (b) Shows that the projection angle $\theta _{0}$ for a projectile launched from the origin is given by

$\theta _{0}=tan^{-1}\left [ \frac{4h_{m}}{R} \right ]$

where the symbols have their usual meaning.

(b)     The maximum height is given by   :                                                                        And,     the horizontal range is given by  :                                                                       Dividing both, we get :                                                                      Hence

Q. 4.32  (a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by   $\theta (t)=tan^{-1}\left [ \frac{v_{0y-gt}}{v_{0x}} \right ]$

where the symbols have their usual meaning.

Using the equation of motion in both horizontal and vertical direction.                                                   and         Now,                                                                                   or                                                              Thus,

Q. 4.31 A cyclist is riding with a speed of  $\inline 27\; km/h.$  As he approaches a circular turn on the road of radius $\inline 80\; m,$ he applies brakes and reduces his speed at the constant rate of  $\inline 0.50 \; m/s$  every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn?

Speed of cycle =   27 Km/h  =  7.5 m/s The situation is shown in figure :-                                 The centripetal acceleration is given by :                                                                                                                                                     And the tangential acceleration is given as  . So, the net acceleration becomes :                  ...

Q. 4.30 A fighter plane flying horizontally at an altitude of $1.5\; km$ with speed  $720\; km/h$  passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed $600 \; m \; \; s ^{-1}$  to hit the plane ? At what minimum altitude should the pilot fly the plane to avoid being hit? $(Take\; g=10\; m\; s ^{-2}).$

According to the question the situation is shown below:-                                               Now,  The horizontal distance travelled by the shell  =  Distance travelled by plane or                                          or                                               or                                                              So,                                                 ...

Q. 4.29 A bullet fired at an angle of $30^{\circ}$ with the horizontal hits the ground $3.0\; km$ away. By adjusting its angle of projection, can one hope to hit a target $5.0\; km$ away? Assume the muzzle speed to be fixed, and neglect air resistance.

The range of bullet is given to be:-       R = 3 Km.                                                            or                                                        or                                                         Now, we will find the maximum range (maximum range occurs when the angle of projection is 450).                                                              or         ...

Q. 4.28 Can you associate vectors with (c) a sphere? Explain.

No,vector cannot be associated with sphere as direction cannot be associated with sphere any how.

Q. 4.28 Can you associate vectors with (b) a plane area, Explain.

(b) The plane area can be expressed in vector form as direction can be associated as pointing outward or inward (normal to the plane) of the area.

Q. 4.28 Can you associate vectors with (a) the length of a wire bent into a loop, Explain.

No, the length of a wire bent into a loop cannot be expressed in vector form as we have no direction associated with it.

Q. 4.27 A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?

The main condition for a physical quantity to be a vector is that it should the law of vector addition. Also, the vector has both direction and, magnitude but these are not sufficient condition. For e.g. current has both magnitude and direction but is a scalar quantity as it doesn't follow the law of vector addition. Rotation is not a vector on a large basis, as it is measured by an angle which...

Q. 4.26     A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors $a$ and $b$  at different locations in space necessarily have identical physical effects ? Give examples in support of your answer

No, a vector doesn't have a definite location as a vector can be shifted in a plane by maintaining its magnitude and direction. Vector can change with time for e.g. displacement vector. No, two equal vectors at a different location may not have identical physical effects. For e.g., two equal force vectors at a different location may have different torque but when they are applied together the...

Q. 4.25 An aircraft is flying at a height of  $3400\; m$ above the ground. If the angle subtended at a ground observation point by the aircraft positions $10.0\; s$ apart is $30^{\circ},$ what is the speed of the aircraft?

The given situation is shown in the figure:-                                For finding the speed of aircraft we just need to find the distance AC as we are given t = 10 sec. Consider  ABD,                                                                                       or                                        or                                                   or                       ...

Q. 4.24  Read each statement below carefully and state, with reasons and examples, if it is true or false :

A scalar quantity is one that

(a) is conserved in a process

(b) can never take negative values

(c) must be dimensionless

(d) does not vary from one point to another in space (e) has the same value for observers with different orientations of axe

(a) False:-  For e.g. energy is a scalar quantity but is not conserved in inelastic collisions. (b) False:-  For example temperature can take negative values in degree Celsius. (c) False:- Since speed is a scalar quantity but has dimensions. (d) False:-  Gravitational potential varies in space from point to point. (e) True:- Since it doesn't have direction.

Q. 4.23  For any arbitrary motion in space, which of the following relations are true :

(a)  $v_{average}=\left ( 1/2 \right )\left [ v\left ( t_{1} \right )+v\left ( t_{2} \right ) \right ]$

(b)  $v_{average}=\left [ r\left ( t_{2} \right ) -r\left ( t_{1} \right )\right ]/\left ( t_{2}-t_{1} \right )$

(c)  $v(t)=v\left ( 0 \right )+a\; t$

(d)  $v(t)=r\left ( 0 \right )+v\left ( 0 \right )t+\left ( 1/2 \right )a\; t^{2}$

(e)  $a_{average}=\left [ v(t_{2})-v(t_{1}) \right ]/\left ( t_{2}-t_{1} \right )$

(a) False:-  Since it is arbitrary motion so the following relation cannot hold all the arbitrary relations. (b) True:-   This is true as this relation relates displacement with time correctly. (c) False: -  The given equation is valid only in case of uniform acceleration motion. (d) False:-   The given equation is valid only in case of uniform acceleration motion. But this is arbitrary motion...

Q. 4.21  A particle starts from the origin at  $t=0\; s$  with a velocity of  $10.0\; \hat{j}\; m/s$  and moves in the x-y plane with a constant acceleration of $\left ( 8.0\; \hat{i}+2.0\; \hat{j} \right )m\; s^{-2}.$

(b) What is the speed of the particle at the time?

The velocity of particle is given by :                                                              Put  t = 2 sec,  So velocity becomes :                                                                 or                                                            Now,  the magnitude of velocity gives :                                                                                            ...

Q. 4.21 A particle starts from the origin at $t=0\; s$  with a velocity of  $10.0\; \hat{j}\; m/s$ and moves in the x-y plane with a constant acceleration of  $\inline \left ( 8.0\; \hat{i}+2.0\; \hat{j} \right )m\; s^{-2}$.

(a) At what time is the x- coordinate of the particle $\inline 16 \; m?$ What is the y-coordinate of the particle at that time

We are given the velocity of the particle as  . And the acceleration is given as :                                                             So, the velocity due to acceleration will be :                                                                So,                                                          By integrating both sides, or                                                     ...

Q. 4.20 The position of a particle is given by $r=3.0t\; \hat{i}-2.0t^{2}\; \hat{j}+4.0\; \hat{k}\; m$ where  $t$ is in seconds and the coefficients have the proper units for $r$ to be in metres.

(b) What is the magnitude and direction of velocity of the particle at $t=2.0\; s ?$

Put value of time   t  = 2   in the velocity vector as given below :                                                                                                   or                                                                                              or                                                                                                    Thus magnitude of velocity is...

Q. 4.20 The position of a particle is given by $r=3.0t\; \hat{i}-2.0t^{2}\; \hat{j}+4.0\; \hat{k}\; m$ where $t$ is in seconds and the coefficients have the proper units for $r$ to be in metres.

(a) Find the $v$ and $a$ of the particle?

(a)  We are given the position vector     The velocity vector is given by:-     or     Now for acceleration :

Q. 4.19 Read each statement below carefully and state, with reasons, if it is true or false :

(c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector

True:-  In a uniform circular motion, acceleration is radially outward all along the circular path. So in 1 complete revolution, all the vectors are cancelled and the null vector is obtained.

Q. 4.19 Read each statement below carefully and state, with reasons, if it is true or false :

(b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point

True:- Because particle moves on the circumference of the circle, thus at any its direction should be tangential in order to move in a circular orbit.
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