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Object 1 has a mass m₁ and a velocity v₁ = (2.50 m/s )x^. Object 2 has a mass m₂ and a velocity v₂ = (3.00 m/s )y^. The total momentum of these two objects has a magnitude of 20.0 kg⋅m/s and points in a direction 65.0° above the positive x axis.part a find m₁

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Final answer:

To calculate mass m1, use the total momentum magnitude and direction to find its component along the x-axis and divide by object 1's velocity, which is solely in the x-direction.

Step-by-step explanation:

To find m1 given the individual velocities of two objects and the total momentum of the system, we must analyze the momentum components in the x and y directions. Since object 1 has a velocity only in the x-axis (v1 = (2.50 m/s)x') and object 2 has a velocity only in the y-axis (v2 = (3.00 m/s)y'), their momenta are perpendicular to each other. The total momentum forms a right triangle with the sum of momenta in the x and y directions as legs, and using Pythagorean theorem, we can find the magnitudes:

Total Momentum (P)^2 = (m1v1)^2 + (m2v2)^2

Substituting the given total momentum (20.0 kg·m/s) and direction (65.0° above the x-axis), we can solve for m1:

20^2 = (m1 * 2.50)^2 + (m2 * 3.00)^2

Without the value of m2, we simply focus on isolating m1 using the angle:

  • P cos(65°) = m1 * 2.50
  • P sin(65°) = m2 * 3.00

Since we need to find m1, we use the first part:

m1 = P cos(65°) / 2.50

Substitute P = 20.0 kg·m/s:

m1 = 20.0 cos(65°) / 2.50

Now we can compute the value of m1.