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Comprehensive Master Sheet: Motion in 1D
Student Name: ____________________________________ Class: 11th (CBSE/NEET/JEE) Subject: Physics
Section A: Distance, Displacement & Basic Kinematics
1.
An athlete completes three and a half rounds of a circular track of radius $R$. Find the ratio of his total distance covered to his net displacement.
2.
An insect crawls $3 \text{ m}$ East, $4 \text{ m}$ North, and then climbs $12 \text{ m}$ vertically up a pole. What is the magnitude of its net displacement from the starting point?
3.
A wheel of radius $R$ is rolling on a straight horizontal road. What is the magnitude of the displacement of the point of the wheel initially in contact with the ground after half a revolution?
4.
Can an object have a constant speed but a varying velocity? Can an object have a constant velocity but a varying speed? Explain with examples.
5.
A particle moves along a straight line. If its distance travelled is directly proportional to the square of time, what can you say about the nature of its acceleration?
6.
A particle is constrained to move on a straight line path. It returns to the starting point after $10 \text{ s}$. The total distance covered by the particle during this time is $30 \text{ m}$. Which of the following statements about the particle is false? (a) Displacement is zero (b) Average speed is $3 \text{ m/s}$ (c) Displacement is $30 \text{ m}$ (d) Average velocity is zero.
7.
A boy walks to his school at a distance of $6 \text{ km}$ with constant speed of $2.5 \text{ km/h}$ and walks back with a constant speed of $4 \text{ km/h}$. Find his average speed for round trip in $\text{km/h}$.
8.
The numerical ratio of displacement to the distance covered is always: (a) Less than one (b) Equal to one (c) Equal to or less than one (d) Equal to or greater than one.
Section B: Average Speed & Average Velocity (Variable Rates)
9.
A car covers the first half of the distance between two places at $40 \text{ km/h}$ and another half at $60 \text{ km/h}$. Calculate the average speed of the car.
10.
A body covers one-third of the total distance with speed $v_1$ and the remaining two-thirds of the distance with speed $v_2$. Find the average speed of the body.
11.
A body moving in a straight line travels with a velocity $v_1$ for time $t_1$, and with a velocity $v_2$ for time $t_2$. What is its average velocity?
12.
A train travels at a speed of $60 \text{ km/h}$ for $0.52 \text{ h}$, at $30 \text{ km/h}$ for the next $0.24 \text{ h}$ and then at $70 \text{ km/h}$ for the next $0.71 \text{ h}$. What is the average speed of the train?
13.
A particle travels half of its total time of journey with speed $v_1$ and the remaining half time with speed $v_2$. Find its average speed.
14.
A man walks on a straight road from his home to a market $2.5 \text{ km}$ away with a speed of $5 \text{ km/h}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 \text{ km/h}$. What is the magnitude of average velocity and average speed of the man over the interval of time $0$ to $50 \text{ min}$?
15.
A car moves a distance of $200 \text{ m}$. It covers the first half of the distance at speed $40 \text{ km/h}$ and the second half of distance at speed $v$. If the average speed is $48 \text{ km/h}$, find the value of $v$.
16.
A point traversing a straight line moves with a constant velocity $v_1$ for $1/n^{\text{th}}$ of the total distance, then with velocity $v_2$ for the next $1/n^{\text{th}}$ of the distance, and so on. Prove that the average velocity is the harmonic mean of the individual velocities.
Section C: Calculus in Kinematics (Differentiation & Integration)
17.
The position of an object moving along x-axis is given by $x = a + bt^2$ where $a = 8.5 \text{ m}$, $b = 2.5 \text{ m/s}^2$ and $t$ is measured in seconds. What is its velocity at $t = 0 \text{ s}$ and $t = 2.0 \text{ s}$? What is the average velocity between $t = 2.0 \text{ s}$ and $t = 4.0 \text{ s}$?
18.
The motion of a particle along a straight line is described by equation $x = 8 + 12t - t^3$ where $x$ is in meters and $t$ in seconds. Find the retardation of the particle when its velocity becomes zero.
19.
A particle moves along a straight line such that its displacement at any time $t$ is given by $S = (t^3 - 6t^2 + 3t + 4) \text{ m}$. Find the velocity when the acceleration is zero.
20.
The acceleration of a particle is increasing linearly with time $t$ as $bt$. The particle starts from origin with an initial velocity $v_0$. Find the distance travelled by the particle in time $t$.
21.
The velocity of a particle is given by $v = (2t^2 - 4t + 3) \text{ m/s}$. Find the acceleration at $t = 2 \text{ s}$.
22.
The velocity $v$ of a particle depends upon the position $x$ as $v = \beta x^{-2n}$. Find the acceleration of the particle in terms of position $x$.
23.
A particle moves such that its position is given by $x = 3t^2$ and $y = 4t^2$, where $x, y$ are in meters and $t$ in seconds. Find the magnitude of its velocity at $t = 2 \text{ s}$.
24.
The relation between time $t$ and distance $x$ is $t = \alpha x^2 + \beta x$, where $\alpha$ and $\beta$ are constants. Show that the retardation is proportional to the cube of the instantaneous velocity.
25.
An object's velocity is given by $v = t^2 - 4t + 3$. Find the total distance covered by the object in the first 4 seconds of its motion.
26.
The acceleration of a particle is given by $a = -k \sqrt{v}$, where $k$ is a positive constant and $v$ is velocity. If initial velocity is $v_0$, find the time taken for the particle to come to rest.
27.
If velocity of a particle is given by $v = \sqrt{180 - 16x} \text{ m/s}$, what will be its acceleration?
28.
A particle starts from rest and its acceleration varies with time as $a = 2t \text{ m/s}^2$. Find its velocity and position after $3 \text{ s}$ if it starts from the origin.
Section D: Equations of Kinematics (Uniform Acceleration)
29.
A driver takes $0.20 \text{ s}$ to apply the brakes after seeing an obstacle. If he is driving at a speed of $54 \text{ km/h}$ and the brakes cause a deceleration of $6.0 \text{ m/s}^2$, find the total distance travelled by the car before stopping.
30.
A bullet fired into a fixed target loses half of its velocity after penetrating $3 \text{ cm}$. How much further will it penetrate before coming to rest assuming that it faces constant resistance to motion?
31.
A particle starting from rest experiences constant acceleration. It travels a distance $x$ in the first $10 \text{ s}$ and a distance $y$ in the next $10 \text{ s}$. Derive the relation between $x$ and $y$.
32.
Two cars A and B start from rest at the same time and accelerate at $a_1$ and $a_2$ respectively. If they cover the same total distance and the first car reaches its destination $t$ seconds earlier than the second car, finding a maximum velocity $v_1$ and $v_2$ respectively, prove that $v_1 - v_2 = t(a_1 a_2 / (a_1 - a_2))$. (Assume they stop immediately).
33.
A body covers $12 \text{ m}$ in the $2^{\text{nd}}$ second and $20 \text{ m}$ in the $4^{\text{th}}$ second of its motion. How much distance will it cover in the $4$ seconds after the $5^{\text{th}}$ second?
34.
A train of length $L$ crosses a pole with constant acceleration. The front of the train passes the pole with velocity $u$ and the rear of the train passes with velocity $v$. Find the velocity with which the middle point of the train passes the pole.
35.
A car moving with a speed of $40 \text{ km/h}$ can be stopped by applying brakes after at least $2 \text{ m}$. If the same car is moving with a speed of $80 \text{ km/h}$, what is the minimum stopping distance?
36.
A particle covers half of its total distance with speed $v_1$ and the rest half distance with speed $v_2$. Its average speed during the complete journey is $v$. Prove that $2/v = 1/v_1 + 1/v_2$.
37.
An electron travelling with a speed of $5 \times 10^4 \text{ m/s}$ passes into an electric field accelerating it at the rate of $10^{15} \text{ m/s}^2$. How long will it take for the electron to double its speed?
38.
A body starts from rest and moves with uniform acceleration. Which of the following graphs represent its motion: (a) Displacement vs Time is a straight line (b) Velocity vs Time is a parabola (c) Displacement vs Time is a parabola (d) Velocity vs Displacement is a straight line.
39.
A car accelerates from rest at a constant rate $\alpha$ for some time, after which it decelerates at a constant rate $\beta$ and comes to rest. If the total time elapsed is $t$, evaluate the maximum velocity acquired and the total distance travelled.
40.
Two trains are moving on the same track in opposite directions at speeds $u_1$ and $u_2$. They are separated by distance $D$ when the drivers apply their brakes, producing retardations $a_1$ and $a_2$. Prove that they will avert a collision if $D > \frac{u_1^2}{2a_1} + \frac{u_2^2}{2a_2}$.
Section E: Motion Under Gravity (Free Fall & Projectiles in 1D)
41.
A ball is thrown vertically upwards with a velocity of $20 \text{ m/s}$ from the top of a multistorey building. The height of the point from where the ball is thrown is $25.0 \text{ m}$ from the ground. How high will the ball rise? How long will it be before the ball hits the ground? ($g = 10 \text{ m/s}^2$)
42.
Two balls are dropped from different heights $h_1$ and $h_2$. Find the ratio of the times taken by them to reach the ground.
43.
A body falls freely from rest. Prove that the distances fallen in successive equal time intervals are in the ratio $1 : 3 : 5 : 7 \dots$ (Galileo's law of odd numbers).
44.
Water drops fall at regular intervals from a tap $5 \text{ m}$ above the ground. The third drop is leaving the tap at the instant the first drop touches the ground. How high above the ground is the second drop at that instant? ($g = 10 \text{ m/s}^2$)
45.
A balloon is ascending at the rate of $9.8 \text{ m/s}$ at a height of $39.2 \text{ m}$ above the ground when a food packet is dropped from it. After how much time and with what velocity does it reach the ground?
46.
A juggler maintains four balls in motion, making each in turn rise to a height of $20 \text{ m}$ from his hand. With what velocity does he project them and where will the other three balls be at the instant when the fourth one is just leaving his hand? ($g = 10 \text{ m/s}^2$)
47.
A stone falls from a tower and travels $45 \text{ m}$ in the last second of its journey. Find the height of the tower. ($g = 10 \text{ m/s}^2$)
48.
A boy drops a stone from a bridge and a second stone $1 \text{ s}$ later. Both stones strike the water simultaneously. If the initial speed of the second stone was $15 \text{ m/s}$, find the height of the bridge. ($g=10 \text{ m/s}^2$).
49.
A ball is dropped from the roof of a tower of height $h$. The total distance covered by it in the last second of its motion is equal to the distance covered by it in the first three seconds. What is the height $h$ of the tower? ($g=10 \text{ m/s}^2$)
50.
A ball is thrown vertically upward with speed $u$. If it is at a certain height $h$ at two different times $t_1$ and $t_2$, prove that $t_1 + t_2 = \frac{2u}{g}$ and $t_1 t_2 = \frac{2h}{g}$.
51.
A parachutist bails out from an aeroplane and after dropping through a distance of $40 \text{ m}$, he opens the parachute and decelerates at $2 \text{ m/s}^2$. If he reaches the ground with a speed of $2 \text{ m/s}$, how long was he in the air? At what height did he bail out?
52.
A body falling freely from a given height $H$ hits an inclined plane in its path at a height $h$. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. For what value of $h / H$ will the body take the maximum time to reach the ground?
Section F: Relative Velocity in 1D & Non-Inertial Frames
53.
Two parallel rail tracks run north-south. Train A moves north with a speed of $54 \text{ km/h}$, and train B moves south with a speed of $90 \text{ km/h}$. What is the velocity of B with respect to A?
54.
From the previous question, find the velocity of a monkey running on the roof of train A against its motion (with a velocity of $18 \text{ km/h}$ with respect to the train A) as observed by a man standing on the ground.
55.
A police van moving on a highway with a speed of $30 \text{ km/h}$ fires a bullet at a thief's car speeding away in the same direction with a speed of $192 \text{ km/h}$. If the muzzle speed of the bullet is $150 \text{ m/s}$, with what speed does the bullet hit the thief's car?
56.
Two trains, each $50 \text{ m}$ long, are travelling in opposite directions with velocities $10 \text{ m/s}$ and $15 \text{ m/s}$. Calculate the time they will take to completely cross each other.
57.
A $120 \text{ m}$ long train is moving towards west with a speed of $10 \text{ m/s}$. A bird flying towards east with a speed of $5 \text{ m/s}$ crosses the train. The time taken by the bird to cross the train will be?
58.
Two cars A and B are moving in the same direction with velocities $30 \text{ m/s}$ and $20 \text{ m/s}$. When car A is $240 \text{ m}$ behind car B, the driver of car A applies the brakes, producing a uniform retardation of $2 \text{ m/s}^2$. Will they collide? If no, find the minimum distance between them.
59.
A passenger is running at maximum speed of $8 \text{ m/s}$ to catch a train. When he is distance $d$ from the door, the train starts from rest with a constant acceleration of $1 \text{ m/s}^2$. What is the maximum value of $d$ so that he can just board the train?
60.
An elevator car is moving upward with uniform acceleration $a$. A passenger drops a coin from a height $h$ above the floor of the elevator. How long will the coin take to strike the floor?
61.
On a two-lane road, car A is travelling with a speed of $36 \text{ km/h}$. Two cars B and C approach car A in opposite directions with a speed of $54 \text{ km/h}$ each. At a certain instant, when the distance AB is equal to AC, both being $1 \text{ km}$, B decides to overtake A before C does. What minimum acceleration of car B is required to avoid an accident?
62.
Ship A is sailing towards the north-east with velocity $\vec{v} = 30\hat{i} + 50\hat{j} \text{ km/hr}$. Ship B is at a distance of $80 \text{ km}$ east and $150 \text{ km}$ north of Ship A and is sailing towards the west at $10 \text{ km/hr}$. A will be at minimum distance from B in what time?
63.
Two particles $P$ and $Q$ move in a straight line $AB$. $P$ starts from $A$ with velocity $u_1$ and acceleration $f_1$. Simultaneously $Q$ starts from $B$ with velocity $u_2$ and acceleration $f_2$ towards $A$. If they pass one another at the middle point of $AB$ and arrive at the other ends of $AB$ with equal velocities, prove that $(u_1 + u_2)(f_1 - f_2) = 8(f_1 u_2 - f_2 u_1)$.
Section G: Graphical Analysis & Curve Interpretations
64.
What does the area under a velocity-time graph and acceleration-time graph represent physically?
65.
The displacement-time graph of a moving particle is a parabola opening upwards. What can you infer about the nature and sign of its acceleration?
66.
A velocity-time graph of an object forms a triangle with the time axis. The base of the triangle is $10 \text{ s}$ and the peak height is $20 \text{ m/s}$. Find the total distance travelled and the maximum acceleration during the journey.
67.
Can a position-time graph have a negative slope? Can it be a vertical straight line? Give physical reasons for your answers.
68.
From a given $v\text{-}t$ graph which consists of a semi-circle strictly above the $t$-axis of radius $2 \text{ m/s}$ from $t=0$ to $t=4 \text{ s}$, find the total displacement of the body.
69.
The graph of $1/v$ versus displacement $x$ is a straight line passing through the origin with a positive slope $k$. Determine the relation between acceleration $a$ and velocity $v$.
70.
An acceleration-displacement ($a-x$) graph for a particle moving in a straight line is given as a straight line passing through $(0, a_0)$ and $(x_0, 0)$. If the particle starts from rest at $x=0$, find its velocity as a function of $x$.
71.
The velocity-displacement ($v-x$) graph of a particle moving in a straight line is a straight line intercepting the $v$-axis at $v_0$ and the $x$-axis at $x_0$. Prove that the acceleration of the particle varies linearly with displacement $x$, and find its maximum magnitude.
72.
A particle's velocity $v$ squared ($v^2$) is plotted against displacement $x$ resulting in a straight line with a positive slope $m$ and y-intercept $C$. What are the physical significances of $m$ and $C$? What is the acceleration of the particle?
73.
A particle starts from rest and undergoes an acceleration $a$ that varies with time $t$ as shown in an $a-t$ graph which is an isosceles triangle with base on the time axis from $t=0$ to $t=2t_0$ and peak $a_0$ at $t=t_0$. Calculate the maximum velocity and total distance covered by the particle in time $2t_0$.
74.
If the position of a particle varies with time as $x = A \sin(\omega t)$, state the nature of the $v-x$ graph. Prove your answer mathematically.
75.
The velocity-time graph of a particle in one-dimensional motion is a straight line passing through the origin. Is the motion uniformly accelerated? If the line does not pass through the origin but has a negative slope, what does it signify about the motion?
Master Answer Key (Short Form)
Q1: $7\pi : 2$
Q2: $13 \text{ m}$
Q3: $R\sqrt{\pi^2+4}$
Q4: Yes(UCM); No
Q5: Constant
Q6: (c) Displacement is 30m
Q7: $3.07 \text{ km/h}$
Q8: (c) $\le 1$
Q9: $48 \text{ km/h}$
Q10: $\frac{3v_1v_2}{2v_1+v_2}$
Q11: $\frac{v_1t_1+v_2t_2}{t_1+t_2}$
Q12: $59.9 \text{ km/h}$
Q13: $\frac{v_1+v_2}{2}$
Q14: $0; 6 \text{ km/h}$
Q15: $60 \text{ km/h}$
Q16: Proof
Q17: $0, 10, 15 \text{ m/s}$
Q18: $-12 \text{ m/s}^2$
Q19: $-9 \text{ m/s}$
Q20: $v_0 t + \frac{bt^3}{6}$
Q21: $4 \text{ m/s}^2$
Q22: $-2n\beta^2 x^{-4n-1}$
Q23: $20 \text{ m/s}$
Q24: Proof
Q25: $4 \text{ m}$
Q26: $\frac{2\sqrt{v_0}}{k}$
Q27: $-8 \text{ m/s}^2$
Q28: $9 \text{ m/s}, 9 \text{ m}$
Q29: $21.75 \text{ m}$
Q30: $1 \text{ cm}$
Q31: $y = 3x$
Q32: Proof
Q33: $136 \text{ m}$
Q34: $\sqrt{\frac{u^2+v^2}{2}}$
Q35: $8 \text{ m}$
Q36: Proof
Q37: $5 \times 10^{-11} \text{ s}$
Q38: (c)
Q39: $\frac{\alpha\beta t}{\alpha+\beta}, \frac{\alpha\beta t^2}{2(\alpha+\beta)}$
Q40: Proof
Q41: $20 \text{ m}; 5 \text{ s}$
Q42: $\sqrt{h_1} : \sqrt{h_2}$
Q43: Proof
Q44: $3.75 \text{ m}$
Q45: $4 \text{ s}, 29.4 \text{ m/s}$
Q46: $20 \text{m/s}, 15,20,15\text{m}$
Q47: $125 \text{ m}$
Q48: $11.25 \text{ m}$
Q49: $125 \text{ m}$
Q50: Proof
Q51: $16.93\text{s}, 234\text{m}$
Q52: $h/H = 1/2$
Q53: $40 \text{ m/s}$ South
Q54: $10 \text{ m/s}$ North
Q55: $105 \text{ m/s}$
Q56: $4 \text{ s}$
Q57: $8 \text{ s}$
Q58: No, $215 \text{ m}$
Q59: $d \le 32 \text{ m}$
Q60: $\sqrt{2h/(g+a)}$
Q61: $1 \text{ m/s}^2$
Q62: $2.6 \text{ hours}$
Q63: Proof
Q64: Disp; Change in v
Q65: Constant, Positive
Q66: $100 \text{ m}$
Q67: Yes(-v); No(inf speed)
Q68: $2\pi \text{ m}$
Q69: $a = -k v^3$
Q70: $v = \sqrt{a_0 x(2-x/x_0)}$
Q71: $a_{max} = v_0^2/x_0$
Q72: $2a, u^2, a=m/2$
Q73: $a_0t_0, a_0t_0^2$
Q74: Ellipse
Q75: Yes; Retardation
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