Energy stored in the mechanical system of linear motion type

The correct formula for calculating the energy stored in a mechanical system of linear motion type is option B, which is given as 1/2 mv^2.

Explanation:

1. Understanding mechanical energy:
Mechanical energy refers to the energy possessed by an object due to its motion or position. In the case of linear motion, mechanical energy is associated with the movement of an object along a straight line.

2. Types of mechanical energy:
There are two forms of mechanical energy: kinetic energy and potential energy.
- Kinetic energy (KE) is the energy possessed by an object due to its motion. It depends on the mass (m) and velocity (v) of the object.
- Potential energy (PE) is the energy possessed by an object due to its position or height. It depends on the mass (m) of the object and the height (h) at which it is located.

3. Energy stored in the mechanical system of linear motion:
In the case of linear motion, the energy stored in the mechanical system is solely kinetic energy since there is no change in height or position.
The formula to calculate the kinetic energy is given by KE = 1/2 mv^2, where:
- KE is the kinetic energy of the object.
- m is the mass of the object.
- v is the velocity of the object.

4. Derivation of the formula:
To understand why the formula for energy stored in the mechanical system of linear motion is 1/2 mv^2, let's consider the following steps:

- The work done (W) on an object is given by the product of the force applied (F) and the displacement (d) along the direction of the force, i.e., W = Fd.
- According to Newton's second law of motion, force (F) is given by the product of mass (m) and acceleration (a), i.e., F = ma.
- Substituting the value of force (F) in the work equation, we get W = mad.
- The equation for acceleration (a) is given by the change in velocity (Δv) divided by the change in time (Δt), i.e., a = Δv/Δt.
- Substituting the value of acceleration (a) in the work equation, we get W = m(Δv/Δt)d.
- Rearranging the equation, we get W = mΔv(d/Δt).
- Since velocity (v) is the change in displacement (Δd) divided by the change in time (Δt), i.e., v = Δd/Δt, we can substitute the value of velocity (v) in the work equation, giving W = m(v)(d/Δt).
- The term (d/Δt) represents the average velocity (v_avg) of the object.
- Therefore, the work done (W) can be written as W = m(v)(v_avg).
- The work done (W) is equal to the energy stored in the mechanical system.
- Thus, the energy stored in the mechanical system is given by E = m(v)(v_avg).
- Since v_avg is half of the final velocity (v