Derive the equation for the kinetic energy of an object of mass m’ moving with velocity v.
The Correct Answer and Explanation is:
To derive the equation for the kinetic energy of an object of mass m moving with velocity v, we start from the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
Step 1: Definition of Work
Work (W) is defined as the force (F) applied on an object times the distance (d) over which the force is applied. Mathematically:W=F⋅dW = F \cdot dW=F⋅d
Now, consider a scenario where an object starts from rest and moves under the influence of a constant force. The force is related to the object’s acceleration (a) by Newton’s second law:F=m⋅aF = m \cdot aF=m⋅a
where m is the mass of the object and a is the acceleration.
Step 2: Relating Velocity and Acceleration
Since the object is accelerating, we can use the kinematic equation that relates velocity and acceleration:v2=u2+2adv^2 = u^2 + 2adv2=u2+2ad
where v is the final velocity, u is the initial velocity (which is 0 in this case, as we are considering motion from rest), a is acceleration, and d is the distance traveled. Substituting u = 0, the equation simplifies to:v2=2adv^2 = 2adv2=2ad
Now, solving for d, we get:d=v22ad = \frac{v^2}{2a}d=2av2
Step 3: Work Done
Now substitute this expression for d into the formula for work:W=F⋅d=(m⋅a)⋅v22aW = F \cdot d = (m \cdot a) \cdot \frac{v^2}{2a}W=F⋅d=(m⋅a)⋅2av2
The acceleration terms cancel out, and we get:W=12mv2W = \frac{1}{2} m v^2W=21mv2
Step 4: Kinetic Energy
Since the work done on the object is equal to the change in its kinetic energy, we conclude that the kinetic energy (KE) of an object of mass m moving with velocity v is:KE=12mv2KE = \frac{1}{2} m v^2KE=21mv2
This is the equation for kinetic energy. It shows that the kinetic energy of an object depends on its mass and the square of its velocity.
