A 55-kg person walks due north with a speed of 1.1 m/s, and her 7.0-kg dog runs directly toward her, moving due south, with a speed of 1.8 m/s.

initial momentum north = 55*1.1-7*1.8

Now I assume she picks the dog up and the system moves on with the same momentum

final momentum north = (55+7)v
so
62 v = 55*1.1 - 7*1.8

To solve this problem, we need to calculate the total momentum before and after the motion.

Momentum is calculated using the formula:
Momentum = mass × velocity

Let's assume the initial velocity of the person is positive (due north) and the velocity of the dog is negative (due south). We can write their initial momenta as:

Person's momentum (initial) = mass of person × velocity of person
= 55 kg × 1.1 m/s

Dog's momentum (initial) = mass of dog × velocity of dog
= 7 kg × (-1.8 m/s)

Now, let's calculate their individual momenta:

Person's momentum (initial) = 55 kg × 1.1 m/s = 60.5 kg·m/s (due north)
Dog's momentum (initial) = 7 kg × (-1.8 m/s) = -12.6 kg·m/s (due south)

The negative sign for the dog's momentum indicates that it's moving in the opposite direction.

To find the total momentum before the motion, we can add the individual momenta:

Total momentum (initial) = Person's momentum (initial) + Dog's momentum (initial)
= 60.5 kg·m/s + (-12.6 kg·m/s)

Now, let's calculate the total momentum:

Total momentum (initial) = 60.5 kg·m/s - 12.6 kg·m/s
= 47.9 kg·m/s (due north)

After the motion, the person and the dog are moving towards each other. To find their final velocity, we use the law of conservation of momentum, which states that the total momentum before the motion is equal to the total momentum after the motion.

Total momentum (initial) = Total momentum (final)

Let's assume the final velocity of the person is Vp (due south) and the final velocity of the dog is Vd (due north). We can write their final momenta as:

Person's momentum (final) = mass of person × velocity of person (final)
= 55 kg × Vp

Dog's momentum (final) = mass of dog × velocity of dog (final)
= 7 kg × Vd

Total momentum (final) = Person's momentum (final) + Dog's momentum (final)

Since the total momentum before and after the motion is the same, we have:

Total momentum (initial) = Total momentum (final)
47.9 kg·m/s = (55 kg × Vp) + (7 kg × Vd)

Now, we need to solve this equation to find the final velocities of the person and the dog.