Let W1 represent the work required to accelerate an object on a frictionless horizontal surface from rest to a speed of v. if W2 represents the work required to accelerate the same object on the same surface from a speed of v to 2v, then which one of the following is the correct relationship between W1 and W2?
a) W2=W1 b) W2=2W1 c) W2=3W1 d) w2= 4W1 e) W2=5W1
c) The kinetic energy change is
W2 = (3/2) m v^2
= 3 W1
To determine the relationship between W1 and W2, we need to understand the concept of work and its relationship with the object's kinetic energy.
Work (W) is defined as the product of force (F) and displacement (d) in the direction of the force. Mathematically, W = F * d * cos(θ), where θ is the angle between the force and displacement vectors.
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. In this case, since the object is being accelerated, the net work done will be equal to the change in kinetic energy.
Let's analyze the situation step by step:
1) Work required to accelerate the object from rest to a speed of v (W1):
Since the object starts from rest, it has an initial kinetic energy of zero. To accelerate the object, a certain amount of work must be done on it. Therefore, W1 is equal to the change in kinetic energy from 0 to 1/2 * m * v^2 (where m is the mass of the object).
2) Work required to accelerate the object from a speed of v to 2v (W2):
At this point, the object already has some kinetic energy. To increase its speed from v to 2v, additional work needs to be done. This additional work will be equal to the change in kinetic energy from 1/2 * m * v^2 to 1/2 * m * (2v)^2.
To find the relationship between W1 and W2, we can compare the changes in kinetic energy in each scenario.
Change in kinetic energy for W1 = 1/2 * m * v^2 - 0 = 1/2 * m * v^2
Change in kinetic energy for W2 = 1/2 * m * (2v)^2 - 1/2 * m * v^2 = 1/2 * m * 4v^2 - 1/2 * m * v^2 = 1/2 * m * 3v^2
Comparing the changes in kinetic energy:
Change in kinetic energy for W2 = 3 * (Change in kinetic energy for W1)
Hence, we can conclude that the correct relationship between W1 and W2 is:
W2 = 3W1
Therefore, the correct answer is option c) W2 = 3W1.