You are pushing a large box across a frictionless floor by applying a constant horizontal force. If the box starts at rest, you have to do work W 1 in order for the box to travel a distance d in time T . How much work would you have to do, in terms of W 1 , to make the box go the same distance in half the time

You would have to do four times the work W₁ to make the box go the same distance in half the time.

Explanation:

By definition work is:

   (1)

Where:

F: is the force and d: is the distance

On the other hand, the force is given by:              

   (2)

By entering equation (2) into (1):

    (3)  

Also, we can express the acceleration in terms of distance and time by using the following kinematic equation:

   (4)

Where:

: is the final distance

: is the initial distance = 0

: is the initial speed = 0 (the box starts at rest)

t: is the time  

Solving for a:

  (5)  

Initially we have:

t = T₁, W = W₁, = d      

And when the box goes the same distance in half the time:

The acceleration is (from equation 5):

   (6)        

Finally, the work W₂ in terms of W₁ is:

Therefore, you would have to do four times the work W₁ to make the box go the same distance in half the time.

I hope it helps you!    

the independent variable, or the one that is intentionally manipulated is the speed.

the part of the wave he is travelling on is the crest