1. REST AND MOTION :
Motion is broadly classified into 3 categories.
1. Rectilinear and translatory motion.
2. Circular and rotatory motion.
3. Oscillatory and vibratory motion.
1.1 Rectilinear or 1-D Motion
When a particle is moving along a straight line, then its motion is a rectilinear motion.
Parameters of rectilinear motion or translatory motion or plane motion :
(A) Time :
(B) Position or location - It is defined with respect to some reference point (origin) of given frame of reference.
Consider a particle which moves from location ( at time t1) to location (at time t2) as shown in the figure below, following path ACB.
(C) Distance :
The length of the actual path traversed by the particle is termed as its distance.
Distance = length of path ACB.
(D) Displacement :
The change in position vector of the particle for a given time interval is known as its displacement.
For a moving particle in a given interval of time
(E) Average speed and average velocity :
Average speed and average velocity are always defined for a time interval.
For a moving particle in a given interval of time
(F) Instantaneous speed and instantaneous velocity
Instantaneous speed is also defined exactly like average speed i.e. it is equal to the ratio of total distance and time interval, but with one qualification that time interval is extremely (infinitesimally) small. The instantaneous speed is the speed at a particular instant of time and may have entirely different value than that of average speed. Mathematically.
When Δs is the distance travelled in time Δt. As Δt tends to zero, the ratio defining speed becomes finite and equals to the first derivative of the distance. The speed at the moment 't' is is called the instantaneous speed at time 't'.
On the distance - time plot, the speed is equal to the slope of the tangent to the curve at the time instant 't'. Let A and B point on the plot corresponds to the time t and t + Δt during the motion. As Δt approaches zero, the chord AB becomes the tangent AC at A. The slope of the tangent equal ds/dt, which is equal to the intantaneous speed at 't'.
(G) Instantaneous velocity :
Instantaneous velocity is defined exactly like speed. It is equal to the ratio of total displacement and time interval, but with one qualification that time interval is extremely (infinitesimally) small. Thus, instantaneous velocity can be termed as the average velocity at a particular instant of time when Δt tend to zero and may have entirely different value that of average velocity : Mathematically.
As Δt tends to zero, the ratio defining velocity becomes finite and equals to the first derivative of the position vector. The velocity at the moment 't' is called the instantaneous velocity or simply velocity at time 't'.
Ex.1 In 1.0 sec a particle goes from point A to point B moving in a semicircle of radius 1.0 m. The magnitude of average velocity is
(A) 3.14 m/sec
(B) 2.0 m/sec
(C) 1.0 m/sec
Sol. Average velocity =
D = AO + OB
= 1 + 1 = 2m
t = 1 sec (given)
⇒ mg of v of = 2m/sec
Ex.2 A particle moves along a semicircular path of radius R in time t with constant speed. For the particle calculate
(i) distance travelled,
(iii) average speed,
(iv) average velocity,
(i) Distance = length of path of particle = = pR
(ii) Displacement = minimum distance between initial and final point
= AB = 2R
(iii) Average speed, v =
(iv) Average velocity =
Ex.3 A body travels the first half of the total distance with velocity v1 and the second half with velocity v2. Calculate the average velocity :
Sol. Let total distance = 2x. Then
total time taken =
∴ Average speed =
Ex.4 Velocity-time equation of a particle moving in a straight line is, v = (10 + 2t + 3t2)
(a) displacement of particle from the origin of time t = 1 s, if it is given that displacement is 20 m at time t = 0
(b) acceleration-time equation.
Sol. (a) The given equation can be written as,
= (10 + 2t + 3t2) dt
or s = 20 + 12 = 32 m
(b) Acceleration-time equation can be obtained by differentiating the given equation w.r.t. time.
or a = 2 + 6t
Specimen Problem :
(A) When equation of displacement is given and speed to be find out
Ex.5 If displacement is depend on time such that x = 2t –2 then find out average speed upto 4 sec.
Sol. Average speed =
for Total distance
at t = 0 it is at x = – 2
at t = 1 it is at 0 m
at t = 4 it is at 6 m.
Total distance = |– 2| + 0 + 6 = 8 m
Average speed = 8/4 = 2m/sec
(B) When velocity is given as a function of time and distance to be find out
In this type of question first find out at what instant the velocity is zero. If this instant is come in our time limit then distance can be calculated by breaking the integration in two part with modulus
Ex.6 If velocity is depend on time such that v = 4 – 2t. Find out distance travelled by particle from 1 to 3 sec.
Sol. Velocity is zero (4 – 2t = 0) at t = 2 sec
So for distance = 4 – 2t ⇒ dx =
dx = 1 + 1 = 2m
(G-1) When velocity is given as a function of t :
v = f(x) ⇒
Ex.7 If velocity is given by following function V = x2. Then find out relation between x & t (assume x = 1 m at t = 0)
Sol. Relation between v & x is
v = x2
we know that v =
(E) Average and instantaneous acceleration.
When the velocity of a moving object/particle changes with time, we can say that it is accelerated.
= Rate of change of velocity
Acceleration is a vector quantity whose direction is same as that of change in velocity vector. Its SI unit is m/s2.
Ex.8 if a = 2x ; initially particle is at x = 2m and is moving with 3 ms–1. Then find out v at x = 5 m.
Sol. Given a = 2x
(E-2) When acceleration is given as function of velocity
Ex.9 If a is depend on v in a following way. a = v and at t = 0 x = 1m, v = 1m/s.
(a) Find out its velocity at t = 2 sec.
(b) Find out its velocity at x = 3 m
Sol. Given a = v
⇒ ln v = 2 ⇒ v = e2 ms–1
(b) Given a = v
(E-3) When acceleration is given as a function of t.
Ex.10 The acceleration of a particle which is depend on time is given by following function a = 2t + 1 and at time t = 0, x = 1m and ui = 2m/s. Then find out displacement of the particle at t = 3 sec.
Sol. ∴ We know that a = ⇒ = 2t + 1 ⇒dv = (2t + 1) dt
So, xf at t = 3 sec is
So, after t = 3 sec the position of the particle is 20.5m but the displacement of the particle is
= 20.5 – 1 = 19.5 m
(E-4) Constant Acceleration Format
Deduce the following equations for uniformly accelerated motion by using integration technique.
(A) v = u + at (B)
(C) v2 – u2 = 2as (D)
First equation of motion. Acceleration is defined as
or dv = adt ...(1)
When time = 0, velocity = u (say)
When time = t, velocity = v (say)
Integrating equation (1) within the above limits of time and velocity, we get
or v – u = a(t – 0)
or v = u + at ...(2)
Second equation of motion. Velocity is defined as
or ds = v dt = (u + at) dt ...(iii)
When time = 0, displacement travelled = 0
When time = t, displacement travelled = s (say).
Integrating equation (3) within the above limits of time and distance, we get
or s – 0 = u (t – 0) +
or s = ut + ...(4)
Third equation of motion. By the definitions of acceleration and velocity,
or ads = vdv ...(5)
When time = 0, velocity = u, displacement travelled = 0
When time = t, velocity = v, displacement travelled = s
Integrating equation (5) within the above limits of velocity and displacement, we get
or v2 – u2 = 2as ...(6)
Fourth equation of motion. By definition of velocity,
or ds = vdt = (u + at) dt ...(7)
When time = (n – 1) second, displacement travelled= sn – 1 (say).
When time = n second, displacement travelled = sn (say)
Integrating equation (7) within the above limits of time and distance, we get
where snth = sn – sn –1 = displacement in nth second.
Ex.11 A car starts from rest and accelerates uniformly for 20 seconds to a velocity of 72 km h–1. It then runs at constant velocity and finally brought to rest in 200 m with a constant retardation. The total distance covered is 600 m. Find the acceleration, retardation and the total time taken.
Sol. (i) Motion with uniform acceleration
Here, u = 0 ; t1 = 20 sec ; v = 72 × = 20 ms–1
∴ v = u + at1
20 = 0 + a × 20 or a = 1 m s–2
Distance travelled by car in this time (20 sec),
(ii) Motion with uniform velocity.
As given, total distance = 600 m
we have calculated S1 = 200 m (with uniform acceleration)
and S2 = 200 m (with retardation)
∴ Net distance for which body moves with uniform velocity,
S = 600 – S1 – S2
= 600 – 200 – 200 = 200 m
∴ Time taken, t =
∴ Total time of journey, t = (20 + 10 + 20) sec
t = 50 sec
Average velocity =
(iii) Motion with uniform retardation.
For this motion, initial velocity, u = 20 m s–1 and final velocity v = 0 ; S2 = 200 m
Acceleration a' = ?
Using, v2 – u2 = 2 a' S2
(0)2 – (20)2 = 2(a') × 200
a' = – 1 ms–2
Let t' = time for which the body comes to rest.
∴ v = u + a' t'
0 = 20 – 1t'
∴ t' = 20 sec.
C. specimen problem
Ex.12 Find out distance travelled by the block in 10 sec. for a given situation.
Sol. First find out it what instant velocity of block becomes zero.
v = u + at
given : u = 10 m/s, a = – 2m/s2
⇒ 0 = 10 – 2t ⇒ t = 5 sec
So we calculate distance for two time intervals.
For first 5 sec.
S1 = ut – at2 ⇒ S1 = (10)(5) – (2) (5)2 = 25 m
for Next 5 sec (block is travelling towards the starting point)
S2 = ut + at2
u = 0
S2 = × 2 × 25 = 25 m
S = S1 + S2 = 50 m
(D) Reaction time :
When a particular situation demands our immediate action, it takes some time before we really respond. Reaction time is the time a person takes to observe, think and act. For example, if a person is driving and suddenly a boy appears on the road, then the time elapse before he applies the breaks of the car is the reaction time. Reaction time depends on complexity of the situation and on an individual.
One can measure one's reaction time by a simple experiment. Take a rule and ask your friend to drop it vertically through the gap between your thumb and forefinger. As soon as it is dropped, note the time elapsed ts before you catch it and the distance d travelled by the ruler. (In a particular case, y was found to be 21.0 cm. Estimate reaction time).
Sol. As the ruler drops under free fall so u = 0, and g = 9.8 ms–2. The distance travelled d and the reaction time tr are related by
Here d = 21.0 cm = 0.21 m
Definition : Time taken by a driver to react for a situation
Reaction Time of the driver
is Δt = t1 – t0
Total distance covered by the car before stopping = distance covered in uniform motion during to to t1 + distance cover in de-accelerated motion during t1 to t2 =
Total distance =
Ex.13 Assume that a car is able to stop with a retardation of 8 ms–2 and that a driver can react to an emergency in 0.5 sec. Calculate the overall stopping distance of the car for a speed of 60 km–1 of the car.
Sol. Here, u = 60 km h–1
Since the application of brakes takes 0.5 s, before this the car was moving with uniform speed of ms -1
∴ Distance covered in 0.5 sec, with a uniform speed is
Now car begins to move with a retardation of 8ms–2
∴ Distance covered before coming to rest, 2a S2 = v2 – u2
∴ Total (overall) distance = S1 + S2 = 8.33 + 17.36 ⇒ S = 25.69 m
Ex.14 Two buses A and B are at positions 50 m and 100 m from the origin at time t = 0. They start moving in the same direction simultaneously with uniform velocity of 10 ms–1 and 5 ms–1. Determine the time and position at which A overtakes B.
Sol. Here we use equation of motion for constant velocity in Cartesian form.
Given x1 (0) = 50 m, x2 (0) = 100 m,
v1 = 10 ms–1, v2 = 5 ms–1
The positions of the two buses at any instant t are
x1 (t) = x1 (0) + v1t = 50 + 10 t
x2 (t) = x2 (0) + v2t = 100 + 10 t
When A overtakes B,
x1 (t) = x2 (t)
50 + 10t = 100 + 5t or 5t = 50
t = 10 s
x1 (10) = x2 (10) = 150 m
Thus A overtakes B at a position of 150 m from the origin at time t = 10 s.
Ex.15 A bus starts from rest with constant acceleration of 5 ms–2. At the same time a car travelling with a constant velocity of 50 ms–1 overtakes and passes the bus. (i) Find at what distance will the bus overtake the car ? (ii) How fast will the bus be travelling then ?
Sol. (i) Suppose the bus overtakes the car after covering distance s.
When the two meet, time taken t is same.
For car, s = 50 t
∴ or t = 20 s
Hence s = 50 t = 50 × 20 = 1000 m.
(ii) v2 = u2 + 2as = 0 + 2 × 5 × 1000 = 10,000 or v = 100 ms–1
(E) Maximum Separation :
What is the maximum separation between car and scooter ?
Sol. Initially separation between car & scooter increases and then decreases.
Separation between them will be maximum at an instant at which velocity of the car is equal to the velocity of scooter.
Velocity car = Velocity scooter = 40 m/s
from v = u + at
40 = 0 + 4t ⇒ t = 10 sec
The distance travelled by scooter in 10 sec.
is S1 = 10 × 40 = 400 m
The distance travelled by car in 10 sec. is
S2 = ut + = 0 + × 4 × (10)2 = 200 m
So maximum separation = S1 – S2 = 400 – 200 = 200 m