Am 85kg jogger is heading due east at a speed of 2.0m/s. A 55kg jogger is heading 32 degrees north of east at a speed of 3.0m/s. Find the magnitude and direction of the sum of the momenta of the two joggers.

Perform a vector addition of the two momenta.

To find the magnitude and direction of the sum of the momenta of the two joggers, we need to calculate the individual momenta vectors of each jogger and then find their vector sum.

The momentum of an object is defined as the product of its mass and velocity. The momentum of each jogger can be calculated using the formula:

Momentum = mass x velocity

For the first jogger (85 kg), heading due east at a speed of 2.0 m/s:
Momentum1 = mass1 x velocity1 = 85 kg x 2.0 m/s = 170 kg·m/s (due east)

For the second jogger (55 kg), heading 32 degrees north of east at a speed of 3.0 m/s:
Momentum2 = mass2 x velocity2 = 55 kg x 3.0 m/s = 165 kg·m/s (32 degrees north of east)

Now, we can represent these momenta vectors graphically and find their sum using vector addition.

Draw a coordinate system with the positive x-axis representing east and the positive y-axis representing north. The first jogger's momentum vector (170 kg·m/s) lies entirely on the x-axis (due east). The second jogger's momentum vector (165 kg·m/s) makes an angle of 32 degrees north of east.

To find the sum of the momenta, add the x-components and y-components of the vectors separately.

The x-component of the sum is the sum of the x-components of each momentum vector:
Sum of x-components = Momentum1 (x-component) + Momentum2 (x-component)
= 170 kg·m/s + 165 kg·m/s (cos 32°)

The y-component of the sum is the sum of the y-components of each momentum vector:
Sum of y-components = Momentum2 (y-component)
= 165 kg·m/s (sin 32°)

To find the magnitude and direction of the sum of the momenta, use the Pythagorean theorem and the inverse tangent function:

Magnitude of the sum of the momenta = sqrt((Sum of x-components)^2 + (Sum of y-components)^2)

Direction (angle) of the sum of the momenta = atan((Sum of y-components) / (Sum of x-components))

Plug in the values to calculate the magnitude and direction of the sum of the momenta.