Question

Two identical blocks of ice move in opposite directions with equal speed and collide with each other. What will be the minimum speed required to make both the blocks melt completely, if the initial temperatures of the blocks were –8 degree Celsius each? (Specific heat of ice is 2100J/kg/K and Latent heat of fusion of ice is $3.36 \times {10^5}J/kg$

Answer 1

Hint: in this question we will use the relation between kinetic energy and initial velocity. Next we will use this relation to get the expression for latent heat and specific capacity. Now substituting the given values, will give us the required answer. Further we will discuss the basics of kinetic energy, potential energy and speed as well.
Formula used:
$K.{E_{\max }} = \dfrac{1}{2}\mu {v^2}$
$m{u^2} = (ms\Delta \theta + mL) \times 2$

Complete answer:
As we know that the specific heat is defined as the amount of heat per unit mass required to raise the temperature by one degree Celsius.
We will first find the expression for initial velocity u, from the kinetic energy, which is given by:
$K.{E_{\max }} = \dfrac{1}{2}\mu {v^2}$
Substituting the values, we get:
$\eqalign{& K.{E_{\max }} = \dfrac{1}{2}\left( {\dfrac{m}{2}} \right){(2u)^2} \cr
  & \Rightarrow K.{E_{\max }} = m{u^2} \cr} $
Now, as we use relation between initial velocity, latent heat and specific capacity of ice:
$m{u^2} = (ms\Delta \theta + mL) \times 2$
Substituting the given values in above equation, we get:
$\eqalign{& \Rightarrow u = \sqrt {2(2100 \times 8 + 336000)} \cr
  & \therefore u = 840m/s \cr} $
Therefore, we get the required result, which gives the minimum speed required to make both the blocks melt completely i.e., 840m/s

Additional information:
The total energy of a system is the sum of kinetic energy and the potential energy. Also, we know that according to the law of conservation of energy, energy can neither be created nor be destroyed; it can only be transferred from one form to another. Electric energy is converted to heat energy by the use of a water heater.
Potential energy is defined as the energy that is stored in an object due to its position relative to some zero position. An object or a body possesses gravitational potential energy if it is positioned at a height above or below the zero height.
Further, Kinetic energy is defined as the form of energy that an object or a particle has by reason of its motion. Kinetic energy is a property of a moving object or particle.
We also know the basic difference of speed and velocity i.e., speed is the measure of how fast an object can travel, whereas velocity tells us the direction of this speed. Speed is a scalar quantity that means it has only magnitude, whereas velocity is a vector quantity that means it has both magnitude and direction. The S.I unit of velocity is meter per second (m/sec).

Note:
We should remember that, if a particle or body is only in motion but does not have any height, the potential energy in that case will be zero. We also know that the S.I unit of speed is m/sec but the speedometer shows the value in km/h.