# Conservation of momentum contradicts the concept of potential energy [duplicate]

The point of potential energy is to say that the amount of KE in an object doesn't resemble all it's potential to do work(like a battery, or gasoline they both have ability to do work even though their momentum is 0).

This contradicts the concept of conservation if momentum, which says that the amount of potential to do work, is always there as speed and mass.

Here are some examples.

When you hold a ball above the ground it has Some PE. When you drop the ball, it loses PE and gains KE, increasing the ball's momentum. Clearly the momentum wasn't conserved.

When a driver hits the gas pedal in his car, it starts accelerating, turning PE of the fuel to KE of the car, increasing the car's momentum. Before he hit the gas pedal, the car had 0 momentum (relative to earth). The momentum wasn't conserved.

There are thousands of examples like those.

Where did my intuition go wrong?

## marked as duplicate by sammy gerbil, stafusa, peterh, Yashas, John Rennie newtonian-mechanics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 1 '17 at 6:22

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## 3 Answers

Your mistake is that you do not look at the whole system. Momentum is only conserved in isolated systems, there can't be any external forces. When the ball falls down, the momentum is gains comes from the earth, because the earth is also attracted by the ball (actio = reactio). The same thing in your second example: The momentum the car gains is taken from the earth, so it starts rotating slowly in the other direction.

This contradicts the concept of conservation if momentum, which says that the amount of potential to do work, is always there as speed and mass.

This is not true. Potential energy is energy stored in the system. There is energy, but something is preventing it from going into more "active" forms of energy, like kinetic energy or heat. Generally it's some energy barrier that prevents the reaction from occurring on it's own (by reaction, I'm including an object falling from a height due to the force of gravity when no longer supported); or it's a transitory state where energy goes between forms of potential energy and more apparent energy.

Conservation of momentum requires a closed system. In your examples, those objects are not closed systems. The closed system that accounts for all the momentum that gives us that potential would be extremely complicated. For fuel, it's a very long process of turning energy of some form into the chemical energy of the fuel.

Closed systems are idealized states for when there are no external interactions. In practice, this never fully happens, we are sometimes just able to ignore the contributions of the bigger system and treat it as though it is isolated.

Your intuition is wrong in that you have neglected to define the system you are dealing with.

System - Ball alone
External force - gravitational attractive force Momentum of ball not conserved because an external force acts on the ball.
Work done by force due to Earth is equal to the kinetic energy gained by the ball.

System = Ball & Earth
External force - none
Momentum of the ball and Earth conserved. As the ball falls down the Earth moves up to meet the ball. Motion of the Earth is not noticeable because the mass of the Earth is so much greater than the ball.
Decrease in gravitational potential energy of the Earth and ball system is equal to the gain in kinetic energy of the ball and the Earth. The gain in kinetic energy of the Earth is very much less than the gain in kinetic energy of the ball.

Note that for the ball alone you cannot define a gravitational potential energy.

• How about if you throw a ball in space? Where does the momentum come from?do you get pushed back when you throw a ball in space? – user170541 Sep 29 '17 at 10:55
• @user170541 You and the Earth suffer a recoil carrying an equal amount of momentum that you gave the ball but in the opposite direction. This is how a jet engine works. – Farcher Sep 29 '17 at 10:57