A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct?

(a) x < 0, v < 0, a > 0 (b) x > 0, v < 0, a < 0

(c) x > 0, v < 0, a > 0 (d) x > 0, v > 0, a < 0

(a)

Key concept: The time rate of change of velocity of an object is called acceleration of the object.

It is a vector quantity. Its direction is same as that of change in velocity (Not of the velocity).

In the table: Possible ways of velocity change

When only direction of velocity changes | When only magnitude of velocity changes | When both magnitude and direction of velocity change |

Acceleration perpendicular to velocity | Acceleration parallel or anti parallel to velocity | Acceleration has two componentsâ€”one is perpendicular to velocity and another parallel or anti parallel to velocity |

E.g.: Uniform circular motion | E.g.: Motion under gravity | E.g: Projectile motion |

Here we will take upward direction positive. As. the lift is coming in downward direction, the displacement will be negative. We have to see whether the motion is accelerating or retarding.

We know that due to downward motion displacement will be negative. When the lift reaches 4th floor and is about to stop velocity is decreasing with time, hence motion is retarding in nature. Thus, x < 0; a > 0. Asdisplacementisinnegativedirection, velocity will also be negative, i.e. v < 0.

The motion of lift will be shown like this.

The position-time (x -1) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. Choose the correct entries in the brackets below:

(a) (A/B) lives closer to the school than (B/A).

(b) (A/B) starts from the school earlier than (B/A).

(c) (A/B) walks faster than (B/A).

(d) A and B reach home at the (same/different) time.

(e) (A/B) overtakes (B/A) on the road (once/twice).

The velocity-time graph of a particle in one-dimensional motion is shown below. Which of the following formula are correct for describing the motion of the particle over the time interval from t_{1} to t_{2}?

In one dimensional motion, instantaneous speed v satisfies 0 < v < v_{0}

(a )The displacement in time T must always take non-negative values.

(b) The displacement x in time T satisfies -v_{()}T < x < v_{0}

(c) The acceleration is always a non-negative number.

(d) The motion has no turning points.

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.

(b) with, zero speed may have non-zero velocity.

(c) with constant speed must have zero acceleration,

(d) with positive value of acceleration must be speeding up.

A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct?

(a) x < 0, v < 0, a > 0 (b) x > 0, v < 0, a < 0

(c) x > 0, v < 0, a > 0 (d) x > 0, v > 0, a < 0

In Exercises 3.13 and 3.14, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?

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.

(b) A monkey sitting on top of a man cycling smoothly on a circular track.

(c) A spinning cricket ball that turns sharply on hitting the ground.

(d) A tumbling beaker that has slipped off the edge of table.

A woman starts from her home at 9.00 am, walks with a speed of 5 km h^{-1} on a straight road up to her office 2.5 km away, stays at the office up to 5.00 pm, and returns home by an auto with a speed of 2.5 km h^{-1}. Choose suitable scales and plot the x-t graph of her motion.

A player throws a ball upwards with an initial speed of 29.4 ms^{-1}.

(a) What is the direction of acceleration during the upward motion of the ball?

(b) What are the velocity and acceleration of the ball at the highest point of its motion?

(c) Choose the x = 0 m and t = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of x-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion.

(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).

A player throws a ball upwards with an initial speed of 29.4 ms^{-1}.

(a) What is the direction of acceleration during the upward motion of the ball?

(b) What are the velocity and acceleration of the ball at the highest point of its motion?

(c) Choose the x = 0 m and t = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of x-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion.

(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).

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.

(b) with, zero speed may have non-zero velocity.

(c) with constant speed must have zero acceleration,

(d) with positive value of acceleration must be speeding up.

Figure gives the x-t plot of a particle executing one ¬dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter 14). Give the signs of position, velocity and acceleration variables of the particle at t = 0.3 s, 1.2 s, – 1.2 s.

A ball is dropped and its displacement versus time graph is as shown (Displacement x from ground and all quantities are positive upwards).

(a) Plot qualitatively velocity versus time graph.

(b) Plot qualitatively acceleration versus time graph.

(a) x < 0, v < 0, a > 0 (b) x > 0, v < 0, a < 0

(c) x > 0, v < 0, a > 0 (d) x > 0, v > 0, a < 0

Figure gives the x-t plot of a particle executing one ¬dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter 14). Give the signs of position, velocity and acceleration variables of the particle at t = 0.3 s, 1.2 s, – 1.2 s.

Figure gives a speed-time graph of a particle in motion along a constant direction. Three equal intervals of time are shown. In which interval is the average acceleration greatest in magnitude? In which interval is the average speed greatest? Choosing the positive direction as the constant direction of motion, give the signs of v and a in the three intervals. What are the accelerations at the points A, B, C and D?

A man runs across the roof-top of a tall building and jumps horizontally with the hope of landing on the roof of the next building which is at a lower height than the first. If his speed is 9 m/s, the (horizontal) distance between the two buildings is 10 m and the height difference is 9 m, will he be able to land on the next building? (Take g = 10 m/s^{2})

Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 71 km h^{-1} in the same direction, with A ahead of B. The driver of B decides to overtake A and accelerates by 1 ms^{-1}. If after 50 s, the guard of B just brushes past the driver of A, what was the original distance between them?

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.

(b) A monkey sitting on top of a man cycling smoothly on a circular track.

(c) A spinning cricket ball that turns sharply on hitting the ground.

(d) A tumbling beaker that has slipped off the edge of table.

A uniformly moving cricket ball is turned back by hitting it with a bat for a very short time interval. Show the variation of its acceleration with time (Take acceleration in the backward direction as positive).

The position-time (x -1) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. Choose the correct entries in the brackets below:

(a) (A/B) lives closer to the school than (B/A).

(b) (A/B) starts from the school earlier than (B/A).

(c) (A/B) walks faster than (B/A).

(d) A and B reach home at the (same/different) time.

(e) (A/B) overtakes (B/A) on the road (once/twice).

A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct?

(a) x < 0, v < 0, a > 0 (b) x > 0, v < 0, a < 0

(c) x >0, v <0, a >0 (d) x >0, v >0, a <0

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?

A jet airplane travelling at the speed of 500 km h-1 ejects its products of combustion at the speed of 1500 km h^{-1 }relative to the jet plane. What is the speed of the latter with respect to an observer on the ground?

In Exercises 3.13 and 3.14, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?

In one dimensional motion, instantaneous speed v satisfies 0 < v < v_{0}

(a )The displacement in time T must always take non-negative values.

(b) The displacement x in time T satisfies -v_{()}T < x < v_{0}

(c) The acceleration is always a non-negative number.

(d) The motion has no turning points.

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?

Figure 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.

Figure gives the x-t plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least? Give the sign of average velocity for each interval.

**Look at the graphs (a) to (d) Fig. carefully and state, with reasons, which of these cannot possibly represent one-dimensional motion of a particle.**

A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and requires 1 s. Plot the x-t graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit 13 m away from the start.

Explain clearly, with examples, the distinction between:

(a) Magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval;

(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 gives the x-t plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least? Give the sign of average velocity for each interval.

A vehicle travels half the distance L with speed V_{1} and the other half with speed v_{2}, then its average speed is

Figure 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.

On a long horizontally moving belt (Fig.), a child runs to and fro with n speed 9 km h^{-1} (with respect to the belt) between his father and mother located 50 a part on the moving belt. The belt moves with a speed of 4 km h^{-1} . For an observe a stationary platform outside, what is the

(a) Speed of the child running in the direction of motion of the belt?

(b) Speed of the child running opposite to the direction of motion of the belt?

(c) Time taken by the child in (a) and (b)?

Which of the answers alter if motion is viewed by one of the parents?

An object falling through a fluid is observed to have acceleration given by a = g – bvwhere g= gravitational acceleration and b is constant After a long time of release, it is observed to fall with constant speed. What must be the value of the constant speed?

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 ms^{-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).

Two stones are thrown up simultaneously from the edge of a cliff 200 m high with initial speeds of 15 ms^{-1} and 30 ms^{-1}. Verify that the graph shown in Fig. 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 ms^{-2}. Give the equations for the linear and curved parts of the plot.

Among the four graph shown in the figure there is only one graph for which average velocity over the time interval (0, T) can vanish for a suitably chosen T. Which one is it?

Thinking Process

In this problem, we have to locate the graph which is having same displacement for two timings. When there are two timings for same displacement the corresponding velocities will be in opposite directions.

Two stones are thrown up simultaneously from the edge of a cliff 200 m high with initial speeds of 15 ms^{-1} and 30 ms^{-1}. Verify that the graph shown in Fig. 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 ms^{-2}. Give the equations for the linear and curved parts of the plot.

A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to the walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground,

(a) the direction of motion of the ball changes every 10 seconds.

(b) speed of ball changes every 10 seconds.

(c) average speed of ball over any 20 second interval is fixed.

(d) the acceleration of ball is the same as from the train.

Explain clearly, with examples, the distinction between:

(a) Magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval;

(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],

For the one-dimensional motion, described by x = t – sin t.

(a) x(t) > 0 for all t > 0 (b) v(t) > 0 for all r > 0

(c) a(t) > 0 for all t > 0 (d) v(t) lies between 0 and 2

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?

A vehicle travels half the distance L with speed V_{1} and the other half with speed v_{2}, then its average speed is

Give examples of a one-dimensional motion where

(a) the particle moving along positive x-direction comes to rest periodically and moves forward.

(b) the particle moving along positive x-direction comes to rest periodically and moves backward.