Exploring Comets Part 3
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EXPLORING COMETS, THE SCIENCE OBJECTIVES OF THE DEEP IMPACT MISSION - PART 3

The Mass of Tempel 1

On a voyage of exploration and discovery you don't know what you will find. But you have to keep on the lookout for certain things. In the case of Deep Impact we will try to find out how massive Comet Tempel 1 is. One method to achieve this involves using the ability to find and track the motion of blocks or, more likely, clumps of material within the ejecta cone.

The following is a simplified, yet reasonably complete, explanation of how astronomers will determine the mass of comet Tempel 1 by tracking material thrown from the crater. However, it should be sufficient for an understanding of the physical principles involved.

Our objective is to measure the acceleration (changes in speed and direction) of blocks of material in the ejecta cone and from that compute the mass of the comet.

What is mass, really?

Mass is the amount of matter, material or just plain stuff, if you will, of which something is made.

We need to distinguish between mass and weight. Weight is the force with which the Earth tugs on material. The more mass, the more weight.

Matter has two properties that we will use to find out how much mass there is in Tempel 1. The properties are inertia and gravity.

Inertia.

Inertia is the property of matter that causes it to resist changes in the speed and/or direction of its motion.

For example, when you want to start or speed up your bike, you need to apply a force through the pedals, gears, chain and wheel in order to do that. If you want to slow down, again, you have to apply a force to a wheel through the brake lever, brake wire, brake pads and wheel. The bike's material "wants" to stay going at whatever speed it is going or remain stopped if not moving.

Similarly, when you want to change the direction in which you are going, the bike "wants" to keep on going in the direction it is already going. In order to change direction you need to apply a force by pointing the front wheel in the direction that you want to go. This is the directional aspect of inertia.

Gravity

Gravity is the property of matter that causes objects to attract each other. For example, you and the earth, the moon and the earth, two lead balls....

It's The Law of Inertia

One of the laws of physics that enables us to measure the mass of Tempel 1 is the law of inertia.

 Note: From the rider's perspective, the rider feels they are being pushed into the seat, rather than the seat pushing them. But here we are looking at the situation from the perspective of the body doing the work, the automobile and it is accelerating the rider.

The law of inertia relates the force needed to change a body's speed and/or direction to its mass and to how rapidly the change takes place. The change in speed and/or direction is called acceleration. If you are sitting in a car and the car speeds up rapidly you can feel the seat back pushing you to speed you up along with the car. That force is causing you to accelerate.

Similarly, if the car takes a sharp turn you can feel the door (or perhaps a fellow passenger) pushing against you and causing you to stay in the car. That force is also due to acceleration, it's due to the change in direction of motion.

 Note: The force is the result of the action on the SUV and that force is equal to the mass of the SUV times the acceleration applied to it.

Suppose your SUV has twice the mass of your family sedan. Then it will take twice as much force (applied where the rubber meets the road) to accelerate the SUV, as it will to accelerate the sedan by an equal amount. For instance: it will take twice the force to accelerate the SUV from a standstill to 60 miles per hour in 15 seconds as it will take to accelerate the family sedan to the same speed in the same amount of time.

Physicists state the law that applies to the forces affecting the bike and the SUV in this way: The force needed to cause a given acceleration is proportional the mass of the object or: force equals mass times acceleration.

The Law of Gravity

Before stating the law that governs gravity, you need to know that there is a point associated with an object called the center of mass. When a force is applied to an object, the object behaves as if all of its mass were located at the center of mass. For example: the center of mass of a baseball is at its center, the center of mass of a doughnut is at the center of its hole.

 Note: To calculate the gravitational force between you and the Earth, multiply your mass times the mass of the Earth and divide by the square of the distance between you and the center of the Earth. With an additional multiplicative factor, called G, that' s the force that keeps your feet on the ground!

Now, the law. The attractive force between two objects due to gravity is proportional to the product of the their masses divided by the square of the distance between their centers of mass. (The square of a number is the number times itself.) Furthermore, the force is directed along the line between the centers of masses.

To Measure the Mass of Tempel 1

An ejecta block must obey both of the above laws.

 Note: The ejecta block is equivalent to the rider in the SUV. The shockwave from the impact is equivalent to accelerating the rider who is not wearing a seat belt.

The law of inertia tells us how much force is needed to produce an observed acceleration (change in speed and/or direction). The law of gravitation tells us how much attractive force gravity can provide for an observed separation of objects.

Once the ejecta block leaves the crater it is no longer propelled by the shockwave that ejected it. Gravity is the only force acting on the block and gravity is trying to pull the ejecta block back toward Tempel 1.

 Note: The law of inertia is important when the ejecta blocks are accelerated away from the comet. The law of gravity acts in the opposite direction to pull the ejecta blocks back to the surface of the comet.

We write an equation using the expression for each force, inertia and gravity. After a bit of algebra, we find that the mass of the ejecta block appearing in each equation, conveniently cancels out.

The mass of the comet is then proportional to the acceleration times the square of the distance between the centers of mass. We can get both acceleration and distance by measuring the position of the ejecta block as it moves away from the crater and thus compute the mass of the comet!!!

 Note: We set the value of inertial force equal to the value of gravitational force. The mass of the ejecta block cancels out of the equation, we don't need to know its exact mass. We can measure the distance between a block and the center of mass of the comet, and we can measure the acceleration on the block. The mass of the comet is then the only unknown quantity remaining in the equation and is expressed as the ratio between the acceleration on the block and the square of the distance between the block and the center of mass of the comet. We have to keep the constant, G, in our calculation too.

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