Given $a + b + c + d = 0$, which of the following statements are correct : (c) The magnitude of $a$ can never be greater than the sum of the magnitudes of $b$, $c$, and $d$,

For any arbitrary motion in space, which of the following relations are true : (d) $r (t) = r (0) + v (0) t + (1/2) a t^2$

Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,

A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?

Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.

(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.

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.

A man can swim with a speed of $4.0 km/h$ in still water. How long does he take to cross a river $1.0 km$ wide if the river flows steadily at $3.0 km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank ?

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

The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (b) What is the magnitude and direction of velocity of the particle at $t = 2.0 s$ ?

Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,

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

Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?

Given $a + b + c + d = 0$, which of the following statements are correct : (d) $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of $a$ and $d$, if they are collinear ?

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 ?

Rain is falling vertically with a speed of $30 m s^{-1}$. A woman rides a bicycle with a speed of $10 m s^{-1}$ in the north to south direction. What is the direction in which she should hold her umbrella ?

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 $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (a) At what time is the x- coordinate of the particle $16 m$? What is the y-coordinate of the particle at that time?

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 $g = 10 m s^{-2}$ ).

A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?

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 $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (b) What is the speed of the particle at the time ?