acceleration and velocity || Prove the relation s = ut + 1/2 at2

VELOCITY AND ACCELERATION

# Velocity: The rate of change of displacement is called velocity. 

# Acceleration: The rate of change of velocity is called acceleration. 

#Rest: If the position of an object does not change its position with respect to its surrounding then, the object is said to be at rest.

#Motion: A body is said to be in motion if it change its position with respect to its surrounding.

# Uniform motion: If a body cover equal distance in equal interval of time then the body is said to be in uniform motion. Eg; motion of stars, planet, etc.

#Non-uniform motion: A body is said to be in non-uniform motion if it does not cover equal distance in equal interval of time.

#Average Velocity: The ratio of total displacement covered by a body and the total time taken by it is called average velocity.

#Relative velocity: The relative velocity of a object is defined as the velocity of that body with respect to another body.

# Retardation: Negative acceleration is called retardation.

# Speed: The rate of change of distance is called speed. 

Q1. Speed is scalar quantity, why?
Ans: Speed is a scalar quantity because it has only magnitude but no direction.

Q2. Is velocity scalar quantity or vector quantity? Why?
Ans: Velocity is a vector quantity because it has both magnitude and direction.

# Difference Between:

S.n	Speed	Velocity
1	It is the rate of change of distance.	It is the rate of change of displacement.
2	It is a scalar quantity.	It is a vector quantity
S.n	Scalar quantity	Vector quantity
1	It has only magnitude.	It has both magnitude and direction.
2	Eg; length, area, volume,etc	Eg; velocity, acceleration, etc
S.n	Acceleration	Retardation
1	The rate of change of velocity is called acceleration.	Negative acceleration is called retardation.
2	Initial velocity is less than final velocity	Final velocity is less than the initial velocity.
S.n	Distance	Displacement
1	It is a scalar quantity.	It is a vector quantity.
2	It cannot be zero.	It can be zero.
3	The actual path travelled by an object is called distance	The shortest distance between any two points is called displacement.

S.n	Uniform Motion	Non-uniform Motion
1	A body is said to be in uniform motion if it travels equal distances in equal interval of time.	: A body is said to be in non-uniform motion if it does not cover equal distance in equal interval of time.
2	Acceleration will be zero in uniform motion	Acceleration will not be zero in non-uniform motion
S.n	Velocity	Acceleration
1	The rate of change of displacement is called velocity.	The rate of change of velocity is called acceleration.
2	Its SI unit is m/s	Its SI unit is m/s2

S.n	Uniform Motion	Non-uniform Motion
1	A body is said to be in uniform motion if it travels equal distances in equal interval of time.	: A body is said to be in non-uniform motion if it does not cover equal distance in equal interval of time.
2	Acceleration will be zero in uniform motion	Acceleration will not be zero in non-uniform motion
S.n	Velocity	Acceleration
1	The rate of change of displacement is called velocity.	The rate of change of velocity is called acceleration.
2	Its SI unit is m/s	Its SI unit is m/s2

 

Some Important Questions:

Q1) Write the definition of velocity. A car moving with the velocity of 15m/s comes to rest after 5 sec. Calculate its retardatio.

Ans:

Velocity: The rate of change of displacement is called velocity.

Here,

final velocity (v) = 0 m/s

initial velocity (u) =15 m/s

time (t) = 5 sec.

retardation (a) = ?

We have,

retardation (r) = (u-v)/t

or, (15 - 0)/5

or, 15/5

3 m/s2



Q2) If a vehicle starts to move from rest with acceleration of 0.3 m m/sec2 after 1 minute, what is its final velocity?

Ans:
Here,
acceleration (a) = 0.3 m/sec2

time (t) = 1 minutes = 60 sec.
initial velocity (u) = 0 m/s
final velocity (v) = ?
We have,
acceleration (a) = (v-u)/t
or, 0.3 = (v - 0)/60
or, 0.3 x 60 = v-0
or, 18 = v - 0
or, v = 18 + 0
or, v = 18m/s
Hence final velocity of a vehicle is 18 m/s.

Q3) A car is travelling with the speed of 90 km/h and another car is with the speed of 20 m/s in opposite direction. Calculate their relative velocity.
Ans:
Here,
Velocity of first car (V1) = 90 km/h = (90 x 1000) / (60 x 60) = 90000/3600 = 25 m/s
Velocity of second car (V2) = 20 m /s
here, both the car are in opposite direction.
so,
Relative velocity (V) = V1 + V2
or, 25 m/s + 15 m/s
40 m/s
Hence the relative velocity is 40 m/s.

Q4) What is vector quantity? A bus starts moving from rest, gains the velocity of 20 m/s after 5 sec. Calculate the acceleration of bus.
Ans:
Vector Quantity : Those quantities which have both magnitude and direction are called vector quantities.
Here,
initial velocity (u) = 0 m/s
final velocity (v) = 20 m/s
time (t ) = 5 sec
We have,
acceleration (a) = (v-u)/t
or, (20 - 0)/5
or, 20/5
4 m/s2

Hence the acceleration of a bus is 4 m/s2.

Q5) Prove the relation s = ut + 1/2 at² where symbols have their usual meanings.
Ans:
Suppose,
u = initial velocity
v = final velocity
s = distance covered
t = time taken
a = acceleration produced


then, we know that, average velocity = (u + v) / 2 ........................(i)
but also we have,
distance covered = average velocity x time taken

or s = ((u+v)/2) x t
s = ((u + u + at)/2) x t ............. [since, v = u + at]
s = ((2u + at)/2) x t
s = (2ut + at²)/2
s = 2ut/2 + at²/2
s = ut + 1/2 at².

Hence proved.

Q6) Prove the equation v = u + at
Soln: Suppose a body is moving from point A to point B with initial velocity (u) and final velocity (v). It completes the distance (s) in time (t).

Then, acceleration produced (a) = (v - u)/t
or at = v - u

v = u + at

Q7) Prove the equation of motion, v² = u² + 2as.
Soln:
Suppose a body moves from A to B with initial velocity (u), final velocity (v), distance travelled is (s) in time (t).
Then, v = u + at
Squaring both the sides,
v² = (u + at)²
v² = u² + 2aut + a²t²
v² = u² + 2a (ut + 1/2at²)
v² = u² + 2as ....................... (since, s = ut + 1/2at²)
Hence Proved.