To find the work done in accelerating an object from rest to a given velocity, you can use the concept of work-energy theorem. The work done (W) is equal to the change in kinetic energy (ΔKE) of the object.
The kinetic energy (KE) of an object can be expressed as:
KE = (1/2) * mass * velocity^2
Assuming the object starts from rest, its initial velocity (v_i) is zero. So, the initial kinetic energy (KE_i) is also zero.
The final kinetic energy (KE_f) when the object reaches the desired velocity (v_f) is:
KE_f = (1/2) * mass * v_f^2
Therefore, the change in kinetic energy (ΔKE) is:
ΔKE = KE_f - KE_i = (1/2) * mass * v_f^2 - 0 = (1/2) * mass * v_f^2
According to the work-energy theorem, this change in kinetic energy is equal to the work done (W):
W = ΔKE = (1/2) * mass * v_f^2
So, the work done in accelerating an object from rest to a given velocity is given by:
W = (1/2) * mass * v_f^2
It's important to note that this equation assumes no other external forces or factors (such as friction) are acting on the object during the acceleration process.