How Do You Calculate the Combined Momentum of Two Joggers?

[CaptFormal]

Homework Statement

An 81.8 kg jogger is heading due east at a speed of 1.81 m/s. A 54.4 kg jogger is heading 32.2° north of east at a speed of 3.00 m/s. Calculate the magnitude of the sum of the momenta of the two joggers.

Homework Equations

m1v1 + m2v2 = m1v1' + m2v2'

The Attempt at a Solution

Since they want the sum of the momenta I figured it meant that I would just add up the masses and the velocities as follows:

(81.8)(1.81) + (54.4)(3.00)

However, that was not the correct answer. Also the angle is throwing me off a bit as I don't see how it could fit into this problem. Any help will be greatly appreciated. Thanks.


CaptFormal

 

[mgb_phys]

Draw a diagram
Remember the momentum in Y (north-south) and X (east-west) is conserved.

 

[tomcue]

dehyde's

Hello,

I would like to clarify that the calculation in the attempt at a solution is incorrect. The equation m1v1 + m2v2 = m1v1' + m2v2' is not applicable in this situation as it only applies to a single object undergoing a change in velocity.

To calculate the magnitude of the sum of the momenta of the two joggers, we must first calculate the individual momenta of each jogger using the formula p = mv, where p is the momentum, m is the mass, and v is the velocity. We can then add the momenta vectorially to find the magnitude of the sum.

For the first jogger, we have:

p1 = (81.8 kg)(1.81 m/s) = 148.358 kg*m/s due east

For the second jogger, we can break down the velocity into its x and y components using trigonometry:

vx = (3.00 m/s)cos(32.2°) = 2.527 m/s due east

vy = (3.00 m/s)sin(32.2°) = 1.621 m/s due north

Therefore, the momentum of the second jogger is:

p2 = (54.4 kg)(2.527 m/s) + (54.4 kg)(1.621 m/s) = 173.906 kg*m/s

To find the magnitude of the sum of the momenta, we can use the Pythagorean theorem:

|p1 + p2| = √[(148.358 kg*m/s)^2 + (173.906 kg*m/s)^2] = 234.214 kg*m/s

Therefore, the magnitude of the sum of the momenta of the two joggers is 234.214 kg*m/s.

The angle is not necessary for this calculation as we are only concerned with the magnitude of the sum of the momenta, not its direction. However, if we wanted to find the direction of the sum of the momenta, we could use the inverse tangent function to find the angle between the resultant momentum vector and the east direction.

I hope this helps clarify the solution. Keep up the good work in your studies!