The vector position of a 3.80 g particle moving in the xy plane varies in time according to the following equation.
r1=(3i+3j)t+2jt^2
At the same time, the vector position of a 5.35 g particle varies according to the following equation.
r2=3i-2it^2-6jt
For each equation, t is in s and r is in cm. Solve the following when t = 2.60
(a) Find the vector position of the center of mass.(i+j)
Use your equations to get the vector locations of each particle at t = 2.60.
Call then (X1,Y1) and (X2,Y2) (or X1i + XY1j and X2i + Y2j))
The X and Y coordinates of the CM will be the mass-weighted mean location:
Xcm = (M1 X1 + M2 X2)/(M1 + M2)
etc.
To find the vector position of the center of mass, we need to calculate the weighted average of the individual particles' positions. The weight for each particle is given by its mass.
Let's start by finding the position of each particle at t = 2.60.
For particle 1:
r1 = (3i + 3j)t + 2jt^2
Substituting t = 2.60:
r1 = (3i + 3j)(2.60) + 2j(2.60)^2
Calculating this expression gives us the position of particle 1 at t = 2.60.
Now, let's do the same for particle 2:
r2 = 3i - 2it^2 - 6jt
Substituting t = 2.60:
r2 = 3i - 2i(2.60)^2 - 6j(2.60)
Calculating this expression gives us the position of particle 2 at t = 2.60.
Next, we need to calculate the center of mass position by finding the weighted average of the positions of the two particles. The weight for each particle is given by its mass.
Let's assume the mass of particle 1 is m1 = 3.80 g and the mass of particle 2 is m2 = 5.35 g.
The center of mass position (cm) is given by:
cm = (m1 * r1 + m2 * r2) / (m1 + m2)
Substituting the positions of the particles we found earlier, we can calculate the center of mass position at t = 2.60.