The formula for kinetic energy is derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.
Let's consider a particle of mass m moving with a velocity v. The work done on the particle is given by the force applied on it multiplied by the distance it moves in the direction of the force. In this case, the force is assumed to be constant.
The work done (W) is given by:
W = F * d
Now, we know that force (F) is equal to mass (m) multiplied by acceleration (a). In this case, the acceleration is the change in velocity (Δv) divided by the time taken (Δt).
F = m * a = m * (Δv / Δt)
Substituting this into the work equation, we get:
W = m * (Δv / Δt) * d
Since velocity (v) is equal to the change in displacement (Δd) divided by the change in time (Δt), we can rewrite the equation as:
W = m * (Δv / Δt) * d = m * (Δd / Δt) * (Δv / Δd) * d
Simplifying further, we get:
W = m * v * Δd
Now, according to the work-energy theorem, the work done on the particle is equal to the change in its kinetic energy (ΔKE).
W = ΔKE
Therefore, we can write:
ΔKE = m * v * Δd
Since ΔKE represents the change in kinetic energy, we can rewrite it as:
KE - KE₀ = m * v * (d - d₀)
Where KE is the final kinetic energy, KE₀ is the initial kinetic energy, d is the final displacement, and d₀ is the initial displacement.
Rearranging the equation, we get:
KE = KE₀ + m * v * (d - d₀)
This is the formula for kinetic energy, which states that the kinetic energy of an object is equal to its initial kinetic energy plus the work done on it, which is the product of its mass, velocity, and the change in displacement.
Therefore, the formula for kinetic energy is derived from the work-energy theorem.
Answers & Comments
Answer:
The formula for kinetic energy is derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.
Let's consider a particle of mass m moving with a velocity v. The work done on the particle is given by the force applied on it multiplied by the distance it moves in the direction of the force. In this case, the force is assumed to be constant.
The work done (W) is given by:
W = F * d
Now, we know that force (F) is equal to mass (m) multiplied by acceleration (a). In this case, the acceleration is the change in velocity (Δv) divided by the time taken (Δt).
F = m * a = m * (Δv / Δt)
Substituting this into the work equation, we get:
W = m * (Δv / Δt) * d
Since velocity (v) is equal to the change in displacement (Δd) divided by the change in time (Δt), we can rewrite the equation as:
W = m * (Δv / Δt) * d = m * (Δd / Δt) * (Δv / Δd) * d
Simplifying further, we get:
W = m * v * Δd
Now, according to the work-energy theorem, the work done on the particle is equal to the change in its kinetic energy (ΔKE).
W = ΔKE
Therefore, we can write:
ΔKE = m * v * Δd
Since ΔKE represents the change in kinetic energy, we can rewrite it as:
KE - KE₀ = m * v * (d - d₀)
Where KE is the final kinetic energy, KE₀ is the initial kinetic energy, d is the final displacement, and d₀ is the initial displacement.
Rearranging the equation, we get:
KE = KE₀ + m * v * (d - d₀)
This is the formula for kinetic energy, which states that the kinetic energy of an object is equal to its initial kinetic energy plus the work done on it, which is the product of its mass, velocity, and the change in displacement.
Therefore, the formula for kinetic energy is derived from the work-energy theorem.
Verified answer
Answer:
Let an object of mass m, move with uniform velocity u.
Let us displace it by s, due to constant force F, acting on it.
Work done on the object of mass 'm' is
W = F × s..... (i)
Due to the force, velocity changes to v and the acceleration produced is 'a'. Relationship between v, u, a and s can be given by formula:v² – u² = 2as
s = v² – u²/2a..... (ii)
F = ma..... (iii)
Substituting (ii) and (iii) in (i) we get
W = F × s
= ma × v² – u²/2a
W = ½m(v²–u²)
If u=0,(object starts at rest)
W = ½mv²
Work done=Change in kinetic energy
Ek = ½mv²
Explanation:
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