An 85-kg jogger is heading due east at a speed of 2.0 m/s. A 55-kg jogger is heading 22° north of east at a speed of 2.7 m/s. Find the magnitude and direction of the sum of momenta of the two joggers.
M1V1+M2V2 = 85*2 + 55*2.7[22o] =
170 + 148.5[22o] = 170 + 148.5*Cos22 +
148.5*sin22 = 170+137.7 + 55.63i =
307.7 + 55.63i = 313[10.25o] kg-m/s.
To find the magnitude and direction of the sum of momenta of the two joggers, we need to use vector addition.
1. Start by breaking down the velocities of both joggers into their components.
Jogger A (heading east):
Velocity (vx1, vy1) = (2.0 m/s, 0 m/s)
Jogger B (heading 22° north of east):
Velocity (vx2, vy2) = (2.7 m/s * cos(22°), 2.7 m/s * sin(22°))
2. Compute the momentum of each jogger using the formula p = mv, where p is momentum, m is mass, and v is velocity.
Momentum of jogger A:
p1 = m1 * v1 = (85 kg) * (2.0 m/s) = 170 kg·m/s
Momentum of jogger B:
p2 = m2 * v2 = (55 kg) * (2.7 m/s * cos(22°)) = 127.102 kg·m/s
3. Add the momenta of both joggers to get the total momentum.
Total momentum = p1 + p2
Total momentum = 170 kg·m/s + 127.102 kg·m/s
Total momentum = 297.102 kg·m/s
4. Find the magnitude and direction of the total momentum.
Magnitude of total momentum = sqrt((px)^2 + (py)^2)
where px and py are the x and y components of the total momentum.
Magnitude of total momentum = sqrt((170 kg·m/s)^2 + (127.102 kg·m/s)^2)
Magnitude of total momentum ≈ 214.843 kg·m/s
To find the direction, you can use the arctan function to find the angle between the x-axis and the total momentum vector.
Direction = arctan(py / px)
Direction = arctan((127.102 kg·m/s) / (170 kg·m/s))
Direction ≈ 36.39° north of east
So, the magnitude of the sum of the momenta of the two joggers is approximately 214.843 kg·m/s, and the direction is 36.39° north of east.