What is kinetic energy?
Kinetic energy, notedE_cE
vs
E, start subscript, c, end subscript, is the energy possessed by a body due to its motion relative to a given frame of reference.
To set an object in motion, a force must be applied. And when the point of application of this force moves, we say that the force is working: there is a transfer of energy. The work is responsible for the variation of the kinetic energy of the object which undergoes this force. The kinetic energy , which is expressed in joules (J), depends on the mass and the speed of the object.
Kinetic energy can be transferred from one system to another but can also be transformed into another form of energy. For example, if a moving billiard ball collides with another stationary ball, then part of the kinetic energy of the moving ball will be transmitted to the second ball, which will in turn start moving. It is also possible that a fraction of the kinetic energy of the first ball is dissipated in another form of energy.
How to calculate kinetic energy?
To find the expression of kinetic energy, we use the concept of work. We are particularly interested in the workWWWwith a forceFFFin a simple case where an object of massmmmis moved a distancedddon a horizontal surface under the action of a force parallel to this surface. As we have already seen:
\begin{aligned} W &= F \times d \\ &= m × a × d\end{aligned}
W
=F×d
=m×has×d
What is the kinetic energy?Eh ? I’m already lost.
Thanks to the equations of kinematics, one replaces the acceleration by its expression according to the initial speedv_\mathrm{i}v
I
What is the gravitational potential energy?
v, start subscript, i, end subscript, the final speedv_\mathrm{f}v
f
v, start subscript, f, end subscriptand distanceddd.Which kinematic formula is it?
\begin{aligned} W &= m\times d\times \frac{v_\mathrm{f}^2-v_\mathrm{i}^2}{2d} \\ &= m\times \frac{v_\ mathrm{f}^2-v_\mathrm{i}^2}{2} \\ &= \frac{1}{2}\times m \times v_\mathrm{f}^2 – \frac{1} {2}\times m \times v_\mathrm{i}^2 \end{aligned}
W
=m×d×
2d _
v
f
2
−v
I
2
=m×
2
v
f
2
−v
I
2
=
2
1
×m×v
f
2
−
2
1
×m×v
I
2
So the work of a force on an object is responsible for the change in quantity\dfrac{1}{2}mv^2
2
1
m v
2
start fraction, 1, divided by, 2, end fraction, m, v, squared, a quantity called kinetic energyE_cE
vs
E, start subscript, c, end subscript.
\text{Kinetic energy: } E_c=\frac{1}{2}\times m\times v^2
E
ˊ
cinema energy
e
ˊ
tick: E
vs
=
2
1
×m×v
2
start text, E, with, \’, on top, n, e, r, g, i, e, space, c, i, n, e, with, \’, on top, t, i, q, u , e, space, colon, space, end text, E, start subscript, c, end subscript, equals, start fraction, 1, divided by, 2, end fraction, times, m, times, v, squared
In a Galilean reference frame, for a point object of mass m traversing a path connecting a point A to a point B, the variation in kinetic energy is equal to the sum of the worksWWWexternal and internal forces acting on this object:
W_{ext/int}=\Delta E_cW
e x t / i n t
=ΔE
vs
W, start subscript, e, x, t, slash, i, n, t, end subscript, equals, delta, E, start subscript, c, end subscript
This general result is known as the kinetic energy theorem; it is valid even in the case of forces whose magnitude and direction vary. This theorem and more generally the conservation of energy and the notion of conservative forces are very useful in solving problems in mechanics.
Why is kinetic energy interesting?
There are several interesting points concerning the kinetic energy that one deduces from its expression.
Kinetic energy depends on the square of the speed of the object. So when the object’s speed doubles, its kinetic energy quadruples. For example, a car traveling at 100 km/h has four times the kinetic energy of the same car traveling at 50 km/h; therefore, in the event of an accident, the risk of damage and injury is greater than the simple factor of 2 linked to the gear ratio.
Kinetic energy is greater than or equal to zero. Indeed, velocity can be positive or negative, but squared velocity is positive or zero.
Kinetic energy is not a vector. So a tennis ball thrown to the right with a speed of 5 m/s has exactly the same kinetic energy as a tennis ball thrown downwards with a speed of 5 m/s.
Exercise 1a: Facing a mass elephantm=6~000m=6 0 0 0m, equals, 6, space, 000kg and having a speedv = 10v=1 0v, equals, 10m/s is rather dangerous. At what speed would a cannonball be fired from111kg if it had the same kinetic energy as the elephant? See the answer.
Exercise 1b: What would be the difference in damage produced by the collision of the cannonball or the elephant on the same brick wall? See the answer.
Exercise 2: Hydrazine, a monopropellant used for rocket propulsion, has an energy densityE_dE
d
E, start subscript, d, end subscriptof1{,}6 \dfrac{\text{MJ}}{\text{kg}}1 , 6
kg
GM
1, comma, 6, start fraction, start text, M, J, end text, divided by, start text, k, g, end text, end fraction. Consider a 100 kg rocket (m_fm
f
m, start subscript, f, end subscript) loaded with 1000 kg (m_pm
p
m, start subscript, p, end subscript) of hydrazine. What is the maximum speed that this rocket can reach? For simplicity, it will be considered that the hydrazine is burned very quickly and that the rocket is not subjected to any external force. See the answer.
