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.
For first stone,
x (0) = 200 m, v (0) = 15 ms-1, a = -10 ms-2
x1 (t) = x (0) + v (0) t + 1/2 a t2
x1 (t) = 200 + 15t – 5t2
When the first stone hits the ground, x1 (t) = 0
– 5t2 + 15t+ 200 = 0 On simplification, t = 8 s
For second stone, x (0) = 200 m, v (0) = 30 ms-1, a = -10 ms-2
x1 (t) = 200 + 30t – 5t2
When this stone hits the ground, x1(t) = 0 .-. -5t2 + 30t + 200 = 0
Relative position of second stone w.r.t. first is given by x2 (t) – x1 (t) = 15t
Since there is a linear relationship between x2(t) – x1 (t) and t, therefore the graph is a straight line.
For maximum separation, t = 8 s So maximum separation is 120 m
After 8 second, only the second stone would be in motion. So, the graph is in accordance with the quadratic equation.
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.
In one dimensional motion, instantaneous speed v satisfies 0 < v < v0
(a )The displacement in time T must always take non-negative values.
(b) The displacement x in time T satisfies -v()T < x < v0
(c) The acceleration is always a non-negative number.
(d) The motion has no turning points.
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 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 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 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?
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 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 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.
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?
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 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 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.
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?
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.
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).
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],
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.
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.
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 (a)Magnitude of average velocity, and (b)Average speed of the man over the interval of time (i) 0 to 30 min. (ii) 0 to 50 min. (iii) 0 to 40 min? [Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero!]
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 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 (a)Magnitude of average velocity, and (b)Average speed of the man over the interval of time (i) 0 to 30 min. (ii) 0 to 50 min. (iii) 0 to 40 min? [Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero!]
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 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 n th second (n=1,2,3……..) versus n. what do you expect this plot to be during accelerated motion: a straight line or a parabola?
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 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 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?
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.
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.
A vehicle travels half the distance L with speed V1 and the other half with speed v2, then its average speed is
A ball is dropped from a building of height 45 m. Simultaneously another ball is thrown up with a speed of 40 m/s. Calculate the relative speed of the balls as a function of time.
The velocity-displacement graph of a particle is shown in figure.
(a) Write the relation between v and x.
(b) Obtain the relation between acceleration and displacement and plot it.
At a metro station, a girl walks up a stationary escalator in time t1 If she remains stationary on the escalator, then the escalator takes her up in time t2. The time taken by her to walk up on the moving escalator will be
(a) (tl+t2)/2 (b) t
(b) t1t2 /(t2 – t1)
(c) t1t2 /(t2 +t1)
(d) t1 —t2
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.
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 boy standing on a stationary lift (open from above) throws a ball upwards with the maximum initial speed he can, equal to 49 m s-1. How much time does the ball take to return to his hands? If the lift starts moving up with a uniform speed of 5 m s-1 and the boy again throws the ball up with the maximum speed he can, how long does the ball take to return to his hands?
A monkey climbs up a slippery pole for 3 and subsequently slips for 3 seconds. Its velocity at time t is given by v(t) = 2t(3 – t); 0 < t < 3 and v(t) = -(t – 3)(6 – t) for 3 < t < 6s in m/s. It repeats this cycle till it reaches the height of 20 m.
(a) At what time is its velocity maximum?
(b) At what time is its average velocity maximum?
(c) At what time is its acceleration maximum in magnitude?
(d) How many cycles (counting fractions) are required to reach the top? Sol. We have to calculate time corresponding to maximum velocity. So we first need to find the maximum velocity in this problem. To calculate maximum dv velocity we will use dv/dt=0
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
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?
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?
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],
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).
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.
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?
The displacement of a particle is given by x = (t- 2)2 where x is in metres and t in seconds. The distance covered by the particle in first 4 seconds is
(a) 4 m
(b) 8 m
(c) 12 m
(d) 16 m
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/s2)
