A police van moving on a highway with a speed of $30 km h^{-1}$ fires a bullet at a thief’s car speeding away in the same direction with a speed of $192 km h^{-1}$. If the muzzle speed of the bullet is $150 m s^{-1}$, with what speed does the bullet hit the thief’s car ? (Note: Obtain that speed which is relevant for damaging the thief’s car).

In which of the following examples of motion, can the body be considered approximately a point object: (b) a monkey sitting on top of a man cycling smoothly on a circular track.

Explain clearly, with examples, the distinction between : (b) magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval]. Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only].

Figure 3.21 shows the $x-t$ plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for $t < 0$ and on a parabolic path for $t > 0$ ? If not, suggest a suitable physical context for this graph.

A car moving along a straight highway with speed of $126 km h^{-1}$ is brought to a stop within a distance of $200 m$. What is the retardation of the car (assumed uniform), and how long does it take for the car to stop ?

In which of the following examples of motion, can the body be considered approximately a point object: (a) a railway carriage moving without jerks between two stations.

Two stones are thrown up simultaneously from the edge of a cliff $200 m$ high with initial speeds of $15 m s^{-1}$ and $30 m s^{-1}$. Verify that the graph shown in Fig. 3.27 correctly represents the time variation of the relative position of the second stone with respect to the first. Neglect air resistance and assume that the stones do not rebound after hitting the ground. Take $g = 10 m s^{-2}$. Give the equations for the linear and curved parts of the plot.

A player throws a ball upwards with an initial speed of $29.4 m s^{-1}$. (d) To what height does the ball rise and after how long does the ball return to the player’s hands ? (Take $g = 9.8 m s^{-2}$ and neglect air resistance).

Two trains $A$ and $B$ of length $400 m$ each are moving on two parallel tracks with a uniform speed of $72 km h^{-1}$ in the same direction, with $A$ ahead of $B$. The driver of $B$ decides to overtake $A$ and accelerates by $1 m s^{-2}$. If after $50 s$, the guard of $B$ just brushes past the driver of $A$, what was the original distance between them ?

Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (a) with zero speed at an instant may have non-zero acceleration at that instant

The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28. Obtain the distance traversed by the particle between (a) $t = 0 s$ to $10 s$.

On a two-lane road, car A is travelling with a speed of $36 km h^{-1}$. Two cars $B$ and $C$ approach car $A$ in opposite directions with a speed of $54 km h^{-1}$ each. At a certain instant, when the distance $AB$ is equal to $AC$, both being $1 km$, $B$ decides to overtake $A$ before $C$ does. What minimum acceleration of car $B$ is required to avoid an accident ?

In which of the following examples of motion, can the body be considered approximately a point object: (c) a spinning cricket ball that turns sharply on hitting the ground.

The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (c) $v_{average} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$

A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (b) average speed of the man over the interval of time (ii) 0 to 50 min

On a long horizontally moving belt (Fig. 3.26), a child runs to and fro with a speed $9 km h^{-1}$ (with respect to the belt) between his father and mother located $50 m$ apart on the moving belt. The belt moves with a speed of $4 km h^{-1}$. For an observer on a stationary platform outside, what is the (a) speed of the child running in the direction of motion of the belt ?.

The position-time ($x-t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 3.19. Choose the correct entries in the brackets below ; (d) A and B reach home at the (same/different) time

The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (b) $v(t_2) = v(t_1) + a (t_2 - t_1)$

Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (d) with positive value of acceleration must be speeding up.

A three-wheeler starts from rest, accelerates uniformly with $1 m s^{-2}$ on a straight road for $10 s$, and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second ($n$ = 1,2,3….) versus $n$. What do you expect this plot to be during accelerated motion : a straight line or a parabola ?