MOTION IN A STRAIGHT LINE -

◊ Mechanics

A branch of physics dealing with properties of a moving body, causes of motion, results due to
motion etc. is known as mechanics.

Mechanics divide into two branches.
(1) Kinematics: A branch of mechanics dealing with motion without considering its causes is
known as kinematics.
(2)Dynamics: A branch of mechanics describing motion along with its causes and properties
of a moving body is called dynamics.

Concept of a particle

A point-like object having mass can be considered as a particle.
If the distance between two objects is very large as compared to their dimensions, these objects
can be treated as particles.

◊ Position and frame of reference:-

To locate the position of the particle we need a frame of reference. A convenient way to fix up the frame of reference is to choose three mutually perpendicular axis (x-y-z) axis.

The coordinate (x,y,z) is the position of particle P with respect to the given frame. Add a clock into the frame to measure the time.

$\fn_cm \vec{r}\rightarrow$ It is the position vector of P w.r.t given frame i.e. the position of the particle in vector form.

$\fn_cm \left [ \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \right ]$

from above fig , we can write

$\fn_cm \vec{r}=x\hat{i}+y\hat{j}$

$\fn_cm \left [ r=\sqrt{x^2+y^2} \right ]\;\;\;and\;\;\;\left [ \tan\theta=\frac{y}{x} \right ]$

In polar coordinates $\fn_cm (r,\theta)$ , generally, angle θ is taken anticlockwise from +ve x-axis.

◊ Rest and motion

An object is said to be in motion if it changes its position w.r.t. its surrounding ( frame of reference) with time. And if its position is not changed w.r.t its surrounding with time then the object is at rest. Note that there is no meaning of rest or motion without the viewer. i.e. Rest and motion are relative terms.

Translatory Motion:- In translatory motion, the body moves in such a way that the linear distance covered by each particle of the body is the same during the motion. Its path may be rectilinear or curvilinear.

Here it is important that the body does not change its orientation. (Circular motion without changing its orientation is an example of translatory motion)

Rotatory Motion:- In rotatory motion, each particle moves in a circular path with a common axis of rotation.

◊ Translatory Motion

1. One-dimensional motion (1d motion):- During motion only one coordinate changes with time.

Ex- motion of a train on a straight track.

2. Two-dimensional motion (2d motion/motion in a plane):- During motion only two coordinate changes with time.

Ex- Projectile motion, circular motion, etc

3. Three-dimensional motion( 3d motion):- All coordinates will change during motion. Ex- motion ob birds, motion of kites, etc.

NOTE:-

The path followed by a point object during its motion is called its trajectory. A trajectory is expressed in the equation.
Point object:- ( having no shape and size)- An object whose distance covered is much greater than its own size is taken as a point object. Ex- A car is a point object when it travels from Deoghar to New Delhi.

◊ Distance and Displacement

The actual length of the path covered by a moving body in between two places is called distance. And the shortest path measured by a body in between two places is called its displacement.

Distance

Path dependent
Scalar quantity having magnitude only
Always +ve
S.I unit is meter(m)
Can never decrease

Displacement

Path independent (depends only on initial and final position)
Vector quantity having both magnitudes as well as direction
Can be +ve, -ve, and zero
S.I unit is meter(m)
may be decreased with time passes

NOTE:-

In straight-line motion in one direction magnitude of displacement and distance are the same.
In a round trip, the magnitude of displacement is zero.
For very small distances, the magnitude of displacement and distance are the same.

$\fn_cm For\;A \;to\;C\;\;\;AC=|\vec{AC}|$

◊ Position Vector / Displacement Vector

QUESTIONS RELATED TO DISTANCE/ DISPLACEMENT

Q.1. Find the Distance and displacement to the given point

Q.2. Draw the position vector of the following coordinate. What is its magnitude and direction with the +ve x-axis?

(3,4), (4,3), (4,4), (-3,-3), (1, √3), (√3,1), (-1,√3),(√3,-1), (-√3,-1), (-2,-2)

Q.3. Find the magnitude and direction of the following vectors.

$\fn_cm (a) \;\vec{A}=3\hat{i}+4\hat{j}$ $\fn_cm (b)\;\vec{B}=-2\hat{i}+2\sqrt{3} \;\hat{j}$ $\fn_cm (c)\; \vec{C}=-2\hat{i}-2\hat{j}$ $\fn_cm (d)\;\vec{D}=\sqrt{3}\;\hat{i}-\hat{j}$

Q.4. Draw displacement vector in the following situations, also find their magnitude and direction. We have the following sets of initial and final positions of the particle.

Initial Point Final point

Q.5. A particle starts moving from the origin, moves 10 m in -ve x direction then turns 90º and moves further 10 m along y direction and finally again turns 90º upward and after moving 20 m, stops. Find the position vector of the particle at the end of its journey.

Q.6. A bird flies due east through a distance of 100 m, then heading due north by a distance of 50 m, it flies vertically up through a distance of 20 m. Find the displacement of the bird relative to its initial position.

Q.7. An insect crawls from A to B where B is the center of the rectangular slant face. Find the

(a) Initial and final position vector (b) Displacement of the insect.

◊ Speed

The rate of change of position of an object with time in any direction is called speed. It is equal to the distance divided by the time taken.

It is a scalar quantity and its S.I unit is m/s.

Average speed:- It is the ratio of the total distance covered to the total time taken

$\fn_cm average \;speed=\frac{total \;distance}{total\;time}$

$\fn_cm \left [ v_{avg}=\frac{\Delta s}{\Delta t} \right ]$

Instantaneous Speed (Speed):- The speed at a given instant of time is known as instantaneous speed.

$\fn_cm v=\lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}\;\;\;\;\;\Rightarrow \;\;\;\;\left [ v=\frac{ds}{dt} \right ]$

◊ Velocity

The rate of change of position of an object with time in a fixed direction is called velocity. It is equal to the displacement divided by the time taken.

It is a vector quantity and the direction is the same as along displacement. Its S.I. unit is m/s.

Average velocity:- It is the ratio of the total displacement covered to the total time taken

$\fn_cm average \;velocity=\frac{total \;displacement}{total\;time}$

$\fn_cm \left [ \vec{v}_{avg}=\frac{\Delta \vec{r}}{\Delta t} \right ]$

Instantaneous velocity (velocity):-The velocity at a given instant of time is known as instantaneous velocity.

$\fn_cm \vec{v}=\lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t}\;\;\;\;\;\Rightarrow \;\;\;\;\left [ \vec{v}=\frac{d\vec{r}}{dt} \right ]$

NOTE:-

For straight-line motion in one direction average speed and magnitude of average velocity are the same.
For any type of motion (Straight or Curve) speed and magnitude of velocity at any point are the same.

QUESTIONS RELATED TO THE AVERAGE SPEED & VELOCITY

1. It is 260 km from Deoghar to Ranchi by air and 320 km by road. An aeroplane takes 30 min to go from Deoghar to Ranchi whereas a deluxe bus takes 8 hrs.

a. Find the avg speed of the plane.
b. Find the avg speed of the bus.
c. Find the avg velocity of the plane
d. Find the avg velocity of the bus

2. When a person leaves his home by car, the meter reads 12352 km. when he returned home after two hrs the reading is 12416 km. what is the avg speed and avg velocity of a car?

3. Find the average speed and average velocity of the particle when it moves from A to B in 2 sec in each case. all distances are in meters.

4. A table clock has its minute hand 4cm long. Find the avg velocity of the tip of the minute hand.

a. Between 6:00 a.m. to 6:30 a.m. and b. Between 6:00 a.m to 6:30 p.m

5. A bus between Deoghar to Asansol passed the 100 km, 160 km, and 220 km points at 10:30 a.m., 11:30 a.m. and 1:30 p.m. Find the avg speed of the bus during each of the following intervals.

a. 10:30 a.m to 11:30 a.m
b. 11:30 a.m to 1:30 p.m and
c. 10:30 a.m to 1:30 p.m

6. A man walks from point (2,4) to point (5,8) along some unknown path in 1 sec. Find the average speed and average velocity of the man.

7. A particle starts from x=0 along a straight path, moves 25 m straight, then stops. Then turns around and moves 10 m back and stops. The total time of motion is 10 sec. What is his average speed and average velocity?

QUESTIONS RELATED TO THE INSTANTANEOUS SPEED/VELOCITY

1. A particle is moving along the x-axis. Its position as a function of time is given by

$\fn_cm x=2.1 t^2+2.8$ (where x is in meters and t is in sec)
Determine
a. The displacement of the particle during the time interval from 3sec to 5sec
b. The avg velocity during this interval and
c. The magnitude of instantaneous velocity at t=5sec.

2. The position of an object moving along the x-axis is given by $\fn_cm x=8.5+2.5t^2$ , where a=8.5m, b=2.5m, and t is measured in seconds. What is the velocity at t=0s and t=2 sec? What is the average velocity between t=2s and 4 s?

3. The displacement in meters of a particle moving along the x-axis is given by $\fn_cm x=18t+5t^2$ . Calculate (a) the instantaneous velocity at t=2s and (b) avg velocity between t=2s and t=3s.

4. The displacement x of a particle varies time t as $\fn_cm x=4t^2-15t+25$ . Find the position, and velocity at t=0. When will the velocity of the particle become zero?

5. The distance x of a particle moving on one dimension under the action of a constant force is related to time t by the equation 𝑡=√𝑥+3. where x is in meters and t in sec. Find the displacement of the particle when its velocity is zero

6. A proton moves along the x-axis according to the equation $\fn_cm t=\sqrt{\frac{x-50t}{10}}$ where x is in meters and t in seconds. Calculate the avg velocity of the proton during the 1st 3 sec of its motion.

7. An electron starting from rest has a velocity v is given by v=kt, where k= 2m/s² and t is the time in sec. What will be the distance covered by the electron in 1st 3 second?

8. A particle moves so that its position vector varies with time as $\dpi{120} \fn_cm \vec{r}=A\cos \omega t\;\hat{i}+B\sin \omega t\;\hat{j}$ . Find

a. Initial velocity of the particle
b. Angle between the position vector and velocity vector of the particle at any time and
c. Speed at any instant

9. An object moving with a speed of 6.25 m/s, is decelerated at a rate given by $\fn_cm \frac{\mathrm{d} v}{\mathrm{d} t}=-2.5\sqrt{v}$ . Find the time taken by the object, to come to rest.

◊ Uniform motion in a straight path

In uniform motion, the object covers equal displacement in an equal time interval in a given direction.
In uniform motion, the average velocity and instantaneous velocity at any point are the same.

i.e. $\fn_cm v=\frac{s}{t}\;\;\;\;\;\Rightarrow \;\;\;\;\;\left [ s=vt \right ]$

QUESTIONS RELATED TO THE UNIFORM MOTION

1. A body travels from A to B at 40 m/s and from B to A at 60 m/s. calculate the average speed and avg velocity.

2. A body travels the first half of the total distance with velocity $\fn_cm v_1$ and the second half with velocity $\fn_cm v_2$ . Calculate the avg velocity.

3. A train moves with a speed of 30 km/h in the first 15 min, with another speed of 40 km/h in the next 15 min, and then with a speed of 60 km/h in the last 30 min. calculate the avg speed of the train for this journey.

4. A car travels along a straight line for the first half-time with a speed of 50 km/h and the second half-time with a speed of 60 km/h. find the avg speed of the car.

5. A body traveling along a straight-line travels one-half of the total distance with a velocity $\fn_cm v_0$ . The remaining part of the distance was covered with a velocity $\fn_cm v_1$ for half the time and with velocity $\fn_cm v_2$ for the other half of time. Find the avg velocity over the whole motion.

6. A train 100m long is moving with uniform velocity of 45 km/h. what is the time it will take to cross bridge 1km long.

7. A particle starts moving along x-axis, with constant velocity of 4 m/s. After 2 sec from the start of motion of the 1st particle, another particle starts in the same direction with the same position with constant velocity of 6 m/s. calculate the time at which second particle will catch the first particle.

8. A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km/h. Finding market closed, he instantly turns and walk back home with a speed of 7.5 km/h. what is the
a. Magnitude of avg velocity
b. Avg speed and avg velocity of the man over the interval (a) 0 to 30 min (b) 0 to 50 min and (c) 0 to 40 min.

9. A block moves in a straight line with velocity v for time $\fn_cm t_0$ . Then its velocity becomes 2v for the next $\fn_cm t_0$ time. Finally, its velocity become 3v for time T. If the average velocity during the complete journey was 2.5v, then find T in term of $\fn_cm t_0$ .

10. A car travels one-third of the distance on a straight road with a velocity of 10 km/h, the next one-third with a velocity 20 km/h, and the last one-third with a velocity of 60 km/h. What is the avg velocity of the car for the entire journey.

11. On a 60 km track, a train travels the 1st 30km with a uniform speed of 30km/h. How fast must the train travel the next 30 km so as to avg 40km/h for the entire trip?

12. A table is given below of a particle moving along x-axis. In the table speed of particle at different time interval is shown. Find the avg speed.

t(s) v(m/s)
0-2 2
2-5 3
5-10 4
10-15 2

◊ Acceleration

When we apply acceleration to a body, its velocity will be changed. It is the rate of change in the velocity of an object.

If velocity increases with time, then its acceleration will be +ve (which means velocity and acceleration are in the same direction ). And if velocity decreases with time then its acceleration will be -ve (which means acceleration is opposite to velocity ). Negative acceleration is known as retardation or deceleration.

Average Acceleration

$\fn_cm Average\;Acceleration=\frac{Change\;in\;velocity}{Total\;time\;taken}$

$\fn_cm \left [ \vec{a}_{avg}=\frac{\Delta\vec{v}}{\Delta t} \right ]$

It is a vector quantity and its direction will be in the direction in which the velocity has changed. Its S.I unit is m/s².

Instantaneous Acceleration/ Acceleration

Acceleration at a particular time of motion is known as instantaneous acceleration

$\fn_cm i.e. \;\;\left [ \vec{a}=\frac{d\vec{v}}{dt} \right ]$

Questions related to the acceleration

(1) A car travels in the +ve x-direction at 20 m/s and needs to pass a truck. The velocity of the car is increased to 25 m/s in 2 sec. Find the average acceleration of the car.

(2) If a car moves with a velocity of 36 km/h in the north direction. It turns left without changing its speed in 10 sec. Find the magnitude and direction of average acceleration.

(3) A car is moving along a circular path with a constant speed of 20 m/s. Find the average acceleration in going from.

(a) A to B in 2 sec (b) A to C in 4 sec (c) A to D in 6 sec (d) A to A in 8 sec

(4) The displacement of a particle moving along the x-axis is given by x=18t+15t² (all in S.I) Calculate

(a) The velocity at t=2 sec
(b) Average velocity between t=2 sec and t=3 sec
(c) Average acceleration between t=2 sec and t=3 sec
(c) Instantaneous acceleration

(5) The displacement x of a particle along x- axis is given by x=3+8t+7t². Obtained the velocity and acceleration at t=2 sec.

(6) The position of a particle is given by x=5t³. Find the velocity and acceleration of the particle as a function of time.

(7) The velocity of a particle is given by the equation v=2t²+5 (all are in S.I). Find

(a) The change in velocity of the particle during the time interval 2 sec to 4 sec.
(b) The average acceleration during the same interval and
(c) the acceleration at 4 sec.

(8) The distance travelled by the particle moving along st. path is given by x= 180t+50t², Find

(a) The initial velocity
(b) Velocity at the end of 4 sec and
(c) The acceleration of the particle.

(9) The acceleration of the particle is given by a=3t²+2t+2. If the particle starts with a velocity of 2 m/s, then find the velocity at the end of 2 sec.

(10) Say a=3t², if initial velocity is 2 m/s. Find the velocity as a function of time.

◊ Uniformly Accelerated Motion

An object is said to be moving with uniform acceleration if its velocity changes by an equal amount in equal time intervals, however small these time intervals may be.

NOTE:- For uniformly accelerated motion, average acceleration and instantaneous acceleration at any point must be the same.

Equations for uniformly accelerated motion (Calculus method)

Consider an object moving with uniform acceleration ‘a’ along +ve x-axis. Let initial velocity is ‘u’ and after time ‘t’ its velocity is ‘v’. During this time interval let its displacement is ‘s’.

We know that acceleration is given by

$\fn_cm a=\frac{\mathrm{d} v}{\mathrm{d} t}$ $\fn_cm \therefore \;\;dv=adt$

integrate both sides with a proper limit

$\fn_cm \int_{u}^{v}dv=a\int_{0}^{t}dt$

$\fn_cm \Rightarrow v-u=at$ $\fn_cm \large {\color{Red} \left [ v=u+at \right ]}$

also, velocity at a point is given by

$\fn_cm v=\frac{\mathrm{d} s}{\mathrm{d} t}$ $\fn_cm \therefore ds=vdt$

$\fn_cm \Rightarrow ds=(u+at)dt$

$\fn_cm \Rightarrow ds=udt+atdt$

integrate both sides with a proper limit

$\fn_cm \int_{0}^{s}ds=u\int_{0}^{t}dt+a\int_{0}^{t}tdt$

$\fn_cm \large \Rightarrow {\color{Red} \left [ s=ut+\frac{1}{2}at^2 \right ]}$

we know that

$\fn_cm a=\frac{\mathrm{d} v}{\mathrm{d} t}$ $\fn_cm \therefore a=\frac{\mathrm{d} v}{\mathrm{d} s}.\frac{\mathrm{d} s}{\mathrm{d} t}= v\frac{\mathrm{d} v}{\mathrm{d} s}\;\;\;\;\;(\because v=\frac{\mathrm{d} s}{\mathrm{d} t})$

$\fn_cm \Rightarrow vdv=ads$

integrate both sides with a proper limit

$\fn_cm \int_{u}^{v}vdv=a\int_{0}^{s}ds$

$\fn_cm \frac{v^2-u^2}{2}=as$

$\fn_cm \therefore$ $\fn_cm \large {\color{Red} \left [ v^2=u^2+2as \right ]}$

Distance covered in $\fn_cm n^{th}$ second

The distance traveled in nth second is given by

$\fn_cm S_{n^{th}}=S_n-S_{n-1}$

$\fn_cm S_{n^{th}}=\left ( un+\frac{1}{2}at^2 \right )-\left ( u(n-1)+\frac{1}{2}a(n-1)^2 \right )$

Solving it, we get

$\fn_cm {\color{Red} \left [ S_{n^{th}}=u+\frac{1}{2}a(2n-1) \right ]}$

NOTE:- If retardation in a forward motion is equal to acceleration in a backward motion then the time taken in a forward motion is equal to that in a backward motion.

Questions related to the uniformly accelerated motion

(1) A car accelerates on a straight road from rest to a speed of 180 km/h in 25 sec. Find the distance covered.

(2) A car travels with a uniform velocity of 72 km/h. The driver applies the break and the car comes to rest with a uniform retardation in 10 sec. Find

(a) The retardation.
(b) Velocity of a car after 3 sec and
(c) the distance travels after the break is applied.

(3) A car starts from rest and accelerates uniformly for 10 sec to a velocity 8 m/s. It then runs at a constant velocity and is finally brought to rest in 64m with constant retardation. The total distance covered by the car is 584m. Find the value of acceleration, retardation, and the total time taken.

(4) A car travels at a uniform velocity of 20 m/s for 5 sec. The break is then applied and the car comes to rest in a further 8 sec. Determine

(a) The distance travels in the first 5 sec.
(b) Retardation in the last 8 sec.
(c) Distance travelled by car after the break is applied and
(d) The total distance travelled

(5) Two trains one travelling at 72 km/h and the other at 90 km/h heading towards one another along the same track. When they are 1 km apart, both the drivers simultaneously see the other’s train and apply a break which retard each train at the rate of 1 m/s². Determine whether the trains would collide or not.

(6) A body has a velocity of 5 m/s and a uniform acceleration of 4 m/s². find the distance travelled in the 11th second of its motion.

(7) A body covers 10 m in the 2nd second and 25 m in the 5th second. If the motion is uniformly accelerated. How far will it travel in 7th second?

(8) A body travels 200 m in the 1st two seconds and 220 m in the next four seconds. What will be the velocity at the end of the 7th second from the start?

(9) A bullet travelling with a velocity of 6 m/s penetrates a tree trunk and comes to rest in 0.4 m. Find the time taken during the retardation.

(10) A particle with an initial velocity of 4 m/s having acceleration a=-2 m/s². Find the velocity, displacement, and distance at t=0, 1, 2,3, 4 sec.

(11) A particle moving along a straight line with a constant acceleration of -4 m/s² passes through point A on the line with a velocity of +8 m/s at some moment. Find the distance travelled by the particle in 5 sec after that moment. (26 m)

(12) A man losses 20% of his velocity after running through 108 m. Prove that he can not run more than 192 m further if his retardation is uniform.

(13) A car travelling at 108 km/h has its speed reduced to 36 km/h after travelling a distance of 200 m. Find the retardation and time taken for this process. (2 m/s², 10 sec)

(14) A body covers 10 m in the 2nd second and 25 m in the 5th second of its motion. If the motion is uniform how far will it go in 7th second? (35 m)

(15) A train accelerates from rest for time t1 at a constant rate α and then it retards at the constant rate β for time t2 and comes to rest. find the ratio t1/t2. (β/α)

(16) Two particles A and B start from rest and move for equal time on a straight line. The particle A has an acceleration a for the first half of the total time and 2a for the second half. The particle B has an acceleration 2a for the first half and a for the second half. Which particle has covered larger distance? (B)

◊ Motion under gravity

When an object is released under the action of the earth’s gravity, then there is a change in the velocity of an object. This change is due to the acceleration exerted by the earth on the object. This acceleration is known as the acceleration due to gravity, whose direction is always vertically downward, its value near the earth is 9.8 m/s².

When a body falls freely, its velocity increases and g is taken +ve. when a body is thrown vertically upward, its velocity decreases and g is taken -ve.

NOTE:-

(1) In the vector method

(2) In case of motion under gravity, Time of ascent=time of descent=u/g.

(3)

(4) A body is thrown vertically upward. If air resistance id to be taken in account, then Time of ascent<Time of descent

Questions related to the motion under gravity

(1) A stone is thrown vertically upward with a velocity of 4.9 m/s. calculate

(a) The maximum height reached
(b) The time taken to reach the max height
(c) The velocity with which it returns to the ground and
(d) The time taken to reach the ground.

(2) A stone thrown upward from the top of a tower 85 m high, reaches the ground in 5 sec (g=10 m/s²). Find

(a) The maximum height above the ground
(b) The velocity with which it reaches the ground and
(c) The time taken to reach the maximum height

(3) A particle is projected up with an initial speed u=10 m/s from the top of a building at time t=0. At time t=5 sec the particle strikes the ground. Find the height of the building. (75 m)

(4) From the top of a tower 100 m in height a ball is dropped and at the same time, another ball is projected vertically upward from the ground with a velocity of 25 m/s. Find when and where the two balls will meet.

(5) From point A, 80 m above the ground a particle is projected vertically upward with a velocity of 29.4 m/s. Five seconds later another particle is dropped from point B, 34.3 m vertically below A. Determine when and where one overtakes the other. (1.6 m above the ground )

(6) A tennis ball is dropped on the floor from a height of 10 m. It rebound to a height of 2.5m. If the ball is connected with the floor for 0.01 sec, what is the average acceleration during the contact (take g=10 m/s²)?

(7) A body falling freely under gravity passes two points 30m apart in 1 sec. Find from what point above the upper point it began to fall.

(8) A balloon is ascending at the rate of 14 m/s at a height of 98 m above the ground when the food packet is dropped from the balloon. After how much time and with what velocity does it reach the ground?

(9) A rocket is fired vertically from the ground with a resultant vertical acceleration of 10 m/s². the fuel is finished in 1 minute, and it continues to move up. what is the maximum height reached?

(10) An object is thrown vertically upward from the ground at 30 m/s.

(a) What is the displacement after 4 sec.
(b) What is the velocity after 4 sec.
(c) What is the maximum height attended.
(d) What is the time of flight.

Exercise 1

1. When does a cyclist appear to be stationary with respect to another moving cyclist?
2. Can the earth be regarded as a point object when it is describing its yearly journey around the earth?
3. Can the displacement be greater than the distance travelled by an object? Give reason.
4. Can a particle in 1D motion have zero speed and a non-zero velocity?
5. Can the speed of a body be negative?
6. Can a body have a constant speed and still have a varying velocity?
7. Can a body have a constant velocity and still have a varying velocity?
8. Can a body have zero velocity and still be accelerating?
9. Can an object have an eastward velocity while experiencing a westward acceleration?
10. Can the direction of velocity of an object change, when acceleration is constant?
11. Is it possible for a body to be accelerated without speeding up or slowing down? if so , give an example.
12. Under what condition is the average velocity equal to the instantaneous velocity?
13. Why is the speed, in general, greater than the magnitude of the velocity?
14. Is the direction of acceleration the same as the direction of velocity?
15. Can we use the equation of kinematics to find the height attend by a body projected upward with any velocity?
16. Two balls of different masses are thrown vertically upward near the earth’s surface with the same initial speed. Which one will rise to the greater height?
17. Two balls of different masses are thrown vertically upwards near the earth’s surface at the same speed. Which one will pass through the point of projection in their downward direction with the greater speed?
18. The displacement of a body is given to be proportional to the cube of time elapsed. What is the nature of the acceleration of the body?
19. Is it possible to have a constant rate of change of velocity when velocity changes both in magnitude and direction? If yes, give one example.
20. Can a body be at rest as well as in motion at the same time? Explain.
21. Which of the two – velocity or acceleration, gives the direction of motion of the body? Justify your answer by an example.
22. What does the speedometer of a car measure- Average speed or instantaneous speed
23. What is the numerical ratio of velocity to speed of an object?
24. A ball hits a wall with a velocity of 30 m/s and renounces with the same speed. What is the change in its velocity?
25. Give an example that shows that a negative acceleration can be associated with a speeding-up object.
26. A body travels, with uniform acceleration a1 for time t1 and with uniform acceleration a2 for time t2. What is the average acceleration?
27. When a particle moves with constant velocity, its average velocity, its instantaneous velocity, and its speed are all equal. Comment on this statement.
28. A car travels at a speed of 60 km/hr due north and the other at a speed of 60 km/hr due east. Are the velocities equal? If no, which one is greater? If you find any of the questions irrelevant, explain.
29. If a particle is accelerating, it is either speeding up or speeding down. Do you agree with this statement?
30. Give examples where (a) the velocity of a particle is zero but its acceleration is not zero, (b) the velocity is opposite in direction to the acceleration, and (c) the velocity is perpendicular to the acceleration.
31. A player hits a baseball at some angle. The ball goes high up in space. The player runs and catches the ball before it hits the ground. Which of the two (the player or the ball) has greater displacement?
32. The increase in the speed of a car is proportional to the additional petrol put into the engine. Is it possible to accelerate a car without putting more petrol or less petrol into the engine?
33. Give an example of a motion for which both the acceleration and velocity are negative?
34. Can an object accelerate if its speed is constant?

Exercise 2

1. Can a body exist in a state of absolute rest or of absolute motion? Explain.
2. What is meant by a point object? Give suitable examples.
3. Distinguish between distance and displacement.
4. Define (a) uniform speed (b) uniform velocity (c) instantaneous speed and velocity (d) uniform acceleration and instantaneous acceleration (e) uniform motion (f) uniformly accelerated motion
5. Deduce the following equation for uniformly accelerated motion by using the calculus method

$\fn_cm (a) \; v=u+at\;(b)\;s=ut+\frac{1}{2}at^2 \;(c)\;v^2=u^2+2as\;(d)\;S_{nth}=u+\frac{a}{2}(2n-1)$

6. Discuss the motion under free fall in one dimension, and write also equations for such type of motion.
7. If in the case of motion, displacement is directly proportional to the square of the time elapsed, what do you think about the acceleration i.e. constant or variable? Explain why.
8. An object is in uniform motion along a straight line. What will be the position-time graph for the motion of the object if (a) x’=+ve, v=+ve (b) x’=+ve, v=-ve (c) x’=-ve, v=+ve, and (d) x’=-ve, v=-ve where x’ represents the position of the object at t=0.

9. Show that the time of ascent of a vertically projected body is equal to the time of descent.

◊ Mechanics

Concept of a particle

◊ Position and frame of reference:-

◊ Rest and motion

◊ Translatory Motion

NOTE:-

◊ Distance and Displacement

NOTE:-

◊ Position Vector / Displacement Vector

QUESTIONS RELATED TO DISTANCE/ DISPLACEMENT

◊ Speed

◊ Velocity

NOTE:-

QUESTIONS RELATED TO THE AVERAGE SPEED & VELOCITY

QUESTIONS RELATED TO THE INSTANTANEOUS SPEED/VELOCITY

◊ Uniform motion in a straight path

QUESTIONS RELATED TO THE UNIFORM MOTION

◊ Acceleration

Average Acceleration

Instantaneous Acceleration/ Acceleration

Questions related to the acceleration

◊ Uniformly Accelerated Motion

Equations for uniformly accelerated motion (Calculus method)

Distance covered in second

Questions related to the uniformly accelerated motion

◊ Motion under gravity

NOTE:-

Questions related to the motion under gravity

Exercise 1

Exercise 2

Apart from this, students also solve the NCERT’s questions

Distance covered in $\fn_cm n^{th}$ second