Both a green ball of mass m and a red ball of mass 2m are pushed the same distan

Question

Both a green ball of mass m and a red ball of mass 2m are pushed the same distance, d, with the different forces, F on the green ball and 2 F on the red ball.

If both balls start at rest and the green ball has 400 J of kinetic energy at the end of the distance d, (please show all work and written explanation for each)

How much work was done on the green ball?

How much work was done one the red ball?

What is the final kinetic energy of the red ball?

How much faster, than the red ball, is the gren ball traveling at the end of the distance d?

Explanation / Answer

Let us assume that mass of the green ball is m

Now mass of the red ball is 2m

The distance through which both the balls are moved be d

Now force applied on the green ball be F

Force applied on the red ball be 2F

Kinetic energy of the green ball be K1 = 400 J

Initial velocity of both the balls be u = 0

a) From Work Energy theorem, work done on an object is equal to change in its kinetic energy.

So work done W = 1/2 m v2 - 1/2 mu2

where v is final velocity of the green ball and u is initial velocity

We know that u = 0

Then W = 1/2 mv2 = final kinetic energy of the green ball.

So work done on the green ball is W = 400 J

b) Work done on the red ball Wr = Force × displacement

Wr = 2F × d

= 2 × ( F × d )

F× d is equal to the work done on the green ball since it is displaced by a distance d using a force F

So work done Wr = 2 × 400 J = 800 J

C) Since red ball too started at rest , work done on the ball is equal to its final kinetic energy.

So kinetic energy of the red ball is K2 = 800 J

We can justify this because to move a heavier object through same distance needed more work which is here provided by doubling the force.

d) Now let the final velocity of the green ball be v (already specified )

Take the final velocity of the red ball as V

Let us take the ratio of their kinetic energies

( K1 / K2 ) = ( 1/2 mv2 ) / ( 1/2 × 2m V2)

( 400/ 800 ) = v2/2V2

1/2 = 1/2 × ( v2/ V2)

. V2 = v2

So finally V = v

This means that both the balls will have same speed at a distance d

Explanation : -

Mass is inversely proportional to acceleration.

As the mass of red ball is double the mass of green ball its acceleration has to be half that of green ball

But a force 2F is applied on the red ball.

As force is directly proportional to acceleration, acceleration now doubles with force.

So the acceleration of the red ball will be same as green ball.

So both will have same speed at every instant.

But as mass is twice, kinetic energy of the red ball will be twice that of green ball.

Related Questions

Navigate

• Browse (All)
• Browse B
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.