Question Video: Studying the Collision of Two Moving Bodies in the Same Line in Two Different Cases Mathematics

Two spheres are moving along a straight line. One has mass 𝑚 and is moving at speed 𝑣, whereas the other has a mass of 10 g and is moving at 36 cm/s. If the two spheres were moving in the same direction when they collided, they would coalesce into one body and move at 30 cm/s in the same direction. However, if they were moving in opposite directions, they would coalesce into one body which would move at 6 cm/s in the direction the first sphere had been traveling. Find 𝑚 and 𝑣.

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Video Transcript

Two spheres are moving along a straight line. One has mass 𝑚 and is moving at speed 𝑣, whereas the other has a mass of 10 grams and is moving at 36 centimeters per second. If the two spheres were moving in the same direction when they collided, they would coalesce into one body and move at 30 centimeters per second in the same direction. However, if they were moving in opposite directions, they would coalesce into one body, which would move at six centimeters per second in the direction the first sphere had been traveling. Find 𝑚 and 𝑣.

There are two scenarios in this question, and in both cases, the bodies coalesce. This means that they join together and the collision is inelastic. In the first scenario, the two bodies are moving in the same direction. In this situation, they join together and move at a speed of 30 centimeters per second. In the second scenario, they were originally moving towards each other. And they end up moving in the same direction as the first sphere with a speed of six centimeters per second. In order to answer this question, we will use the conservation of momentum. This states that the momentum before is equal to the momentum after. The formula we will use is 𝑚 one 𝑢 one plus 𝑚 two 𝑢 two is equal to 𝑚 one 𝑣 one plus 𝑚 two 𝑣 two.

We will now clear some space to answer the question. As the spheres are joining together, we will only have one product on the right-hand side. In our first scenario, we have 𝑚𝑣 plus 10 multiplied by 36 is equal to 𝑚 plus 10 multiplied by 30. Distributing the parentheses gives us 30𝑚 plus 300. We can then subtract 360 from both sides so that 𝑚𝑣 is equal to 30𝑚 minus 60. We repeat this for the second scenario. The only differences are the 36 is now negative as the second sphere is moving in the opposite direction and the final velocity is six. This simplifies to 𝑚𝑣 minus 360 is equal to six 𝑚 plus 60. This time, we can add 360 to both sides, such that 𝑚𝑣 is equal to six 𝑚 plus 420.

We now have a pair of simultaneous equations where the left-hand side of both of them is 𝑚𝑣. This means that 30𝑚 minus 60 must be equal to six 𝑚 plus 420. Adding 60 and subtracting six 𝑚 from both sides gives us 24𝑚 is equal to 480. We can then divide both sides by 24 giving us 𝑚 is equal to 20. The mass of the first sphere is 20 grams. We can now substitute this into one of our equations. We will choose equation one. This gives us 20𝑣 is equal to 30 multiplied by 20 minus 60. The right-hand side simplifies to 540. We can then divide both sides of the equation by 20 giving us 𝑣 is equal to 27. The initial velocity of the first sphere is 27 centimeters per second. We have now found the mass and speed of the first sphere that would result in the two spheres coalescing on collision.

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