Class 9th
Work And Energy
chapter 06
Student Learning Objectives
Kinetic Energy | Mathematical formula Derivation| Examples | SLO Questions
The energy associated with the body due to its motion is known as Kinetic energy. When work is done on the body and it starts moving, then work is transferred to body in the form of Kinetic energy.
- Explanation:
When a stone is thrown, it moves and possesses kinetic energy. Also , when a car is moving, it also possesses Kinetic energy. Now consider both the stone and car moving at the same speed. Of course, car possess more energy due to higher mass. Similarly, if two cars are approaching towards us , car with higher speed will possess more energy.
So it is clear that Kinetic energy depends upon mass of body and speed of body . Mathematically, it is half the product of mass and square of velocity.
K.E = 1/2 mv²
- Nature of Kinetic energy:
Kinetic energy is a scaler Quantity because it depends on the mass of body and magnitude of velocity.
- Derivation of formula:
Consider a scenario in which all the work done on a cart is transferred to it in the form of Kinetic energy. Consider a body of mass ‘m’ is at rest , it’s intial Velocity ‘Vi’ is zero. A Horizental force ‘F’ is applied to it and it starts moving with Velocity ‘Vf’ and covers some displacement S .
Work is done on the cart and is given by,
w = F . s …… (i)
From Newton’s 2nd law of motion, this applied force is equal to
F = ma
As no time is involved, the displacement s by Third equation of motion,
2as = Vf² – Vi²
s = Vf² – Vi² / 2a
Since Vi = 0 so Vf = V
s = V² – 0² / 2a
s = V² / 2a
Putting values of F and s in equation (i)
w = (ma) ( V² / 2a)
w = 1/2 (mV²)
This work done is transferred to the cart in the form of Kinetic energy.
So ,
K.E = 1/2 (mV²)
SLO Question:
What will be effect on Kinetic energy of body , if mass of body is doubled?
Answers:
We know that , the kinetic energy of body is given by ,
K.E = 1/2 mv² …….. (i)
Now if mass of body is Doubled , kinetic energy will be changed ,
m’ = 2m
So K.E’ = 1/2 ( m’V²)
Putting value m’ in above equation,
K.E’ = 1/2 ( 2mV²)
K.E’ = 2 ( 1/2 mV² )
K.E’ = 2 K.E
So kinetic energy will be doubled when mass of a body is doubled.
What will be effect on Kinetic energy of body , if velocity of body is tripled?
Answers:
We know that , the kinetic energy of body is given by ,
K.E = 1/2 mv² …….. (i)
Now if velocity of body is tripled , kinetic energy will be changed ,
V’ = 3V
So K.E’ = 1/2 ( mV’²)
Putting value of V’ in above equation,
K.E’ = 1/2 ( m (3V²))
K.E’ = ( 1/2 m9V² )
K.E’ = 9 ( 1/2 mV²)
K.E’ = 9 K.E
So kinetic energy will increase 9 times when velocity of a body is tripled.
Example 6.2
A 60.0 g bullet is fired from a gun with 3150 J of kinetic energy. Find it’s Velocity.
Given Data :
mass of bullet = m = 60.0 g = 60.0/1000 kg = 0.06 kg
Kinetic energy= K.E = 3150 J
Required:
Velocity of bullet = V = ?
Solution:
We know that Kinetic energy is given by ,
K.E = 1/2 mv²
K.E × 2 = mv²
( K.E × 2) / m = v²
v² = (K.E × 2 ) /m
v² = ( 3150 J × 2) / (0.06 kg )
v² = 6300 / 0.06
v² = 105,000 m²/s²
v = √(105,000m²/s²)
v = 324 m/s
The bullet of mass 60 g Having kinetic energy 3150 J will have 324 m/s velocity.
Assignment 6.2
A bullet of mass 30 g travels at a speed of 400m/s. Calculate it’s kinetic energy.
Given Data :
mass of bullet = m = 30 g = 30/1000 kg = 0.03 kg
Velocity of bullet = v = 400m/s
Required:
Kinetic energy of bullet = K.E = ?
Solution:
As we know that ,
K.E = 1/2 mv²
K.E = 1/2 { 0.03 × (400²)}
K.E = 1/2 { 0.03× 160000 }
K.E = 1/2 { 4800}
K.E = 2400 J
The bullet will possess 2400 J of energy if it has mass of 30 g and moves with velocity of 400 m/s.#
What will be effect on Kinetic energy of body , if mass of body is halved?
Answers:
We know that , the kinetic energy of body is given by ,
K.E = 1/2 mv² …….. (i)
Now if mass of body is halved , kinetic energy will be changed ,
m’ = 1/2m
So K.E’ = 1/2 ( m’V²)
Putting value m’ in above equation,
K.E’ = 1/2 ( 1/2 mV²)
K.E’ = 1/2 ( 1/2 mV² )
K.E’ =1/2 K.E
So kinetic energy will be halved when mass of a body is halved.
What will be effect on Kinetic energy of body , if velocity of body is doubled?
Answers:
We know that , the kinetic energy of body is given by ,
K.E = 1/2 mv² …….. (i)
Now if velocity of body is doubled , kinetic energy will be changed ,
V’ = 2V
So K.E’ = 1/2 ( mV’²)
Putting value of V’ in above equation,
K.E’ = 1/2 ( m (2V²))
K.E’ = ( 1/2 m4V² )
K.E’ = 4 ( 1/2 mV²)
K.E’ = 4 K.E
So kinetic energy will increase 4 times when velocity of a body is doubled.
What will be effect on Kinetic energy of body , if velocity of body is halved?
Answers:
We know that , the kinetic energy of body is given by ,
K.E = 1/2 mv² …….. (i)
Now if velocity of body is halved , kinetic energy will be changed ,
V’ = 1/2 V
So K.E’ = 1/2 ( mV’²)
Putting value of V’ in above equation,
K.E’ = 1/2 ( m (1/2V)²)
K.E’ = ( 1/2 m1/4V² )
K.E’ = 1/4 ( 1/2 mV²)
K.E’ = 1/4 K.E
So kinetic energy will decrease 4 times when velocity of a body is halved.
Related Example:
A body moving with velocity 40 m/s has kinetic energy of 4150 Joule. What will be mass of body ?
Given Data:
Velocity of body = v = 40 m/s
Kinetic energy= K.E = 4150 Joule
Required:
Mass of body = m = ?
Solution:
As we know that , K.E = 1/2 ( mv²)
K.E / v² = 1/2 (m)
(2 × K.E) /v² = m
m = (2×K.E)/v²
m = (2 × 4150) / (40²)
m = 8300 / 1600
m = 5.18 kg
