Conservation Laws Conservation of Momentum II

Two Dimensional Explosions Momentum is conserved overall and it is also conserved in both the x and y directions. Use the same equations and techniques that we used in the previous lesson. Only this time solve everything twice. First solve the explosion in the x-direction using only x velocities. Then solve the explosion in the y-direction using only y velocities.

Example 1 A 5 kg mass explodes into three pieces. A 1 kg fragment moves in the -y direction at 6 m/s. A 2 kg fragment moves in the -x direction at 4 m/s. Determine the speed and direction of the third fragment. This splits into three fragments. We need the modify the equations. 5 kg Initially the problem begins with a 5 kg mass. It never tells you what the mass is originally doing. Assume the simplest case. It must be at rest , v0 = 0 . This means v0x = 0 and v0y = 0 Now take a look at what happens as a result of the explosion.

Example 1 A 5 kg mass explodes into three pieces. A 1 kg fragment moves in the -y direction at 6 m/s. A 2 kg fragment moves in the -x direction at 4 m/s. Determine the speed and direction of the third fragment. The second and third sentences tell us part of what happens during the explosion. They are not specific about what happens to the third fragment. 2 kg -4 m/s 2 kg 1 kg -6 m/s Before exploding the total mass was 5 kg. The mass of the third fragment must be 2 kg .

Example 1 Solve the x-direction Solve the y-direction 1 kg -6 m/s 2 kg A 5 kg mass explodes into three pieces. A 1 kg fragment moves in the -y direction at 6 m/s. A 2 kg fragment moves in the -x direction at 4 m/s. Determine the speed and direction of the third fragment. 1 kg -6 m/s 2 kg -4 m/s Solve the x-direction +3 m/s +4 m/s Solve the y-direction

Example 1 A 5 kg mass explodes into three pieces. A 1 kg fragment moves in the -y direction at 6 m/s. A 2 kg fragment moves in the -x direction at 4 m/s. Determine the speed and direction of the third fragment. Now Solve for the velocity and direction of the third fragment. 2 kg +4 m/s +3 m/s 5 m/s 37o Or, you could recognize this one as a 3-4-5 triangle and save some work. The calculations to the left would be necessary if the right triangle wasn’t easy.

Two Dimensional Collisions Momentum is conserved overall and it is also conserved in both the x and y directions. Use the same equations and techniques that we used in the previous lesson. Only this time solve everything twice. First solve the collision in the x-direction using only x velocities. Then solve the collision in the y-direction using only y velocities.

Example 2 First let’s imaging what the motion looks like 5 m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 1 kg 2 kg 5 m/s

Example 2 First let’s imaging what the motion looks like 5 m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 2 kg 5 m/s 1 kg

Example 2 First let’s imaging what the motion looks like 5 m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 2 kg 5 m/s 1 kg

Example 2 First let’s imaging what the motion looks like 5 m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 2 kg 5 m/s 1 kg

Example 2 First let’s imaging what the motion looks like ? m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 2 kg ? m/s 1 kg 5 m/s

Example 2 First let’s imaging what the motion looks like ? m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 2 kg ? m/s 1 kg 5 m/s © RJansen

Example 2 First let’s imaging what the motion looks like ? m/s 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? First let’s imaging what the motion looks like 2 kg ? m/s 1 kg 5 m/s

Example 2 ? m/s First let’s imaging what the motion looks like 2 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? 2 kg ? m/s First let’s imaging what the motion looks like We also only need to look at just the instant before the collision and the instant right after the collision. Why? It was constant velocity before and constant velocity after. 1 kg 5 m/s

Example 2 Let’s move this over to get some room for the equations. A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Let’s move this over to get some room for the equations. 2 kg ? m/s 1 kg 5 m/s

Components Example 2 ? m/s I see vectors at angles. v1y  A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? 2 kg ? m/s 1 kg 5 m/s I see vectors at angles. v1x v1y  What is the secret first step to solve this problem? Components 37o 4 m/s 3 m/s We have a lot of information about the 1 kg mass (mass 2). Solving for the velocity components does not take much effort in this case. The 2 kg mass (mass 1) has unknown components. If we solve for these then we can find the overall velocity and angle of this mass.

Example 2 Solve in the x-direction ? m/s v1y  2 kg v1x 4 m/s 1 kg 37o A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Solve in the x-direction 2 kg ? m/s 1 kg 5 m/s 37o 4 m/s 3 m/s v1x v1y 

Example 2 Solve in the x-direction 2 kg v1x 3 m/s 5 m/s 4 m/s 1 kg A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Solve in the x-direction 2 kg 1 kg 5 m/s 4 m/s v1x 3 m/s

Example 2 Solve in the x-direction Now Solve in the y-direction ? m/s A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Solve in the x-direction 2 kg ? m/s 1 kg 5 m/s 37o 4 m/s 3 m/s v1y  Now Solve in the y-direction

Example 2 Solve in the x-direction Now Solve in the y-direction A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Solve in the x-direction 2 kg 1 kg 3 m/s v1y 1.5 m/s Now Solve in the y-direction Remember, down is negative

Example 2 Solve in the x-direction Now Solve in the y-direction ? m/s A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Solve in the x-direction 1.5 m/s 2 kg ? m/s 1 kg 5 m/s 37o 4 m/s 3 m/s  Now Solve in the y-direction

Example 2 Solve in the x-direction Now Solve for v1 and  A 2 kg mass moving at 5 m/s in the +x-direction strikes a 1 kg ball at rest. The collision is slightly off center. After the collision the 1 kg ball moves with a speed of 5 m/s at an angle of 37o below the x-axis. What is the speed and direction of the 2 kg ball? Solve in the x-direction 2 kg ? m/s 1.5 m/s  3 m/s Now Solve for v1 and  Now Solve in the y-direction