Momentum Equations

Equations are shortcuts that help you get to answers. Take a minute to review the equations for momentum. These are equations that you will use again, so copy them down. 

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This is the basic equation or definition for momentum. Mass times velocity equals momentum. Remember, mass is measured in grams or kilograms, and velocity is measured in (m/s). Also, remember that velocity has direction, so sometimes you might need to use a negative sign to show if an object is going in the opposite direction.

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This equation says that a force applied for an amount of time causes the momentum to change.

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This equation works for a collision where two objects stick together.

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This equation says that the momentum of the two objects before a collision equals the momentum of the two objects after the collision.

 

Figure It Out

Follow your problem solving strategies. Look for what information you are given. Draw a quick sketch to help you think about what is happening and then write down the information from the problem.

 

mass = 70 kg        velocity = 20 m/s        Momentum is what you are looking for.

Which equation puts these three terms together? 

Solve it:  p= (70 kg) x (20 m/s) = 1400 kg·m/s

 

Now try this one. 

How fast would a 0.2 kg football have to be traveling to have the same momentum as a 0.05 kg bullet travelling 400 m/s? There are two objects, so label your information:

football's mass= 0.2kg

football's velocity = ?

bullet's mass =0.05 kg

bullet's velocity = 400 m/s

 

You know the bullet's mass and velocity, so you can multiply them to find the bullet's momentum:

(0.05) x (400) = 20 kg*m/s

 

Now set that momentum equal to the football's mass times velocity:

20 kgm/s = (0.2 kg) x (v)

 

Solve for v --> 100 m/s; the football has to travel at 100 m/s to have the same momentum as the bullet.

 

 

Keep Going!

List the information:

 

mass = 50 kg      initial velocity = 25 m/s and final velocity = 0 m/s, so the change in velocity is 25 m/s.

amount of time= 0.4 seconds    You are looking for force.

Which equation puts all of these together in one equation?   

Solve it:  F x (0.4 s) = (50 kg) x (25 m/s)  (multiply 50 by 25 first and then divide that by 0.4)

Force equals 3,125 Newtons! That's a lot of force!

 

 

Try this one: A big league hitter attacks a fast ball. The ball has a mass of 0.15 kg, and it is pitched at 36 m/s. After the hit, the ball is traveling at 45 m/s in the opposite direction. The impact lasted 0.002 seconds. How big of a force did the hitter put on the ball? List information:

mass = 0.15 kg

initial velocity = 36 m/s

final velocity = 45 m/s in the opposite direction

Here is the biggie! That means the change in the velocity was all the way from +36 to -45 or 81 m/s!

time = 0.002 s

 

F (0.002) = (0.15) (81)  (multiply 0.15 by 81 then divide by 0.002)

F = 6075 Newtons

 

 

Collisions

There are two objects. Make the bullet mass 1 and the wood block mass 2. Look at the two equations for collisions. Which one makes sense here?   

 

 0.02 kg;  = 200 m/s;  = 0.3 kg;  = 0 m/s   Then the other side of the equation has the velocity you are looking for.

Solve it:  (0.02) x ( 200)  + (0.3) x (0)  =  (0.02 + 0.3) v    (First, notice that you have 0.3 multiplied by 0. Whenever something is multiplied by 0, the answer is 0, so that piece of the equation equals 0.)  The equation now is as follows:

(0.02) x (200) = (0.02 +0.3) v.  (Multiply 0.02 by 200 and you get 4;  add 0.02 and 0.3 and you get 0.32.)

4= (0.32) v ;  v= 12.5 m/s (The bullet and the block are now traveling 12.5 m/s.)

 

Try this one. 

Two kids are roller skating. Amy, with a mass of 45 kg, is traveling forward at 3 m/s. Jenny, who has a mass of 50 kg, is traveling in the opposite direction at 5 m/s. They crash into each other and hold onto each other, so that they move as one mass. How fast are they traveling? Make Sarah mass 1 and Jenny mass 2.

   

 45 kg;  = 3 m/s;  = 50 kg;  = -5 m/s   Then the other side of the equation has the velocity you are looking for.

Be careful! Notice that Jenny's velocity has a negative sign because she is going in the opposite direction.

(45) x (3) + (50) x (-5) = (45 + 50) v

135        +   (-250)     = 95 v

        -115 = 95 v   --> v = -1.2 m/s

What does that mean? The girls keep moving in the direction Jenny was going at the beginning at 1.2 m/s together.

 

 

Last Problem

For this problem, make the ball mass 1 and the pin mass 2. Then use this equation because the two objects don't stay connected after the collision: 

 

 

 = 7 kg       =9 m/s      = 2 kg      = 0 m/s

                     = ?                                 = 14 m/s

Put the values in the equation: 

                                                 (7 x 9)   +    (2 x 0)    = (7 x  )     +   (2 x 14)

                                                    63      +     0        =   ( 7 x  )      +  28 

                                                    63 - 28                  = (7 x  )

                                                        35 = 7 x   -->  = 5 m/s

The bowling ball is still going 5 m/s after it hits the pin.

 

Now try this practice problem: John has to hit a bottle with a ball to win a prize. He throws a 0.5 kg ball with a velocity of 20 m/s. It hits a 0.2 kg bottle, and the bottle flies forward at 30 m/s. How fast is the ball traveling after hitting the bottle? In this problem, you will make the ball mass 1 and the bottle mass 2. Use this equation because the two objects don't stay connected after the collision: 

 

 = 0.5 kg      = 20 m/s     = ?         = 0.2kg        = 0 m/s       = 30 m/s

 

Put the values in the equation:

(0.5 x 20) + (0.2 x 0) = (0.5  ) + (0.2 x 30)

    10        +        0       = (0.5  ) +  6

                    10 -6 = (0.5  )

                        4 = (0.5  )

                          = 8 m/s

 

Assignment: Momentum and Collision Math Exploration

It's time to show what you can do with these types of problems. Complete the assignment and turn it in. Use the sample and practice problems to guide you.

 

 

 

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