Judy (mass 45.0 kg), standing on slippery ice, catches her leaping dog, Atti (mass 12kg), moving horizontally at 4.0 m/s. Use the conservation of momentum to find the speed of Judy and her dog after the catch.

klm m

12

According to the law of conservation of momentum, the total momentum before the catch is equal to the total momentum after the catch. In this case, the initial momentum is the product of the mass and velocity of Judy and Atti, and the final momentum is the product of the combined mass and final velocity after the catch.

Let's represent the velocity of Judy after the catch as v1 and the velocity of Atti after the catch as v2.

The initial momentum before the catch is given by:

Initial momentum before = (mass of Judy × velocity of Judy) + (mass of Atti × velocity of Atti)
= (45.0 kg × 0 m/s) + (12 kg × 4.0 m/s)
= 0 + 48 kg·m/s
= 48 kg·m/s

The final momentum after the catch is given by:

Final momentum after = (combined mass of Judy and Atti) × (velocity of Judy and Atti after the catch)
= (45.0 kg + 12 kg) × (v1 + v2)

Since the momentum before and after the catch is the same, we can equate them:

48 kg·m/s = (45.0 kg + 12 kg) × (v1 + v2)

Now, we need to solve for v1 + v2. To do this, we need the combined mass of Judy and Atti, which is:

Mass of Judy and Atti = mass of Judy + mass of Atti
= 45.0 kg + 12 kg
= 57.0 kg

Plugging this value back into the equation:

48 kg·m/s = 57.0 kg × (v1 + v2)

Now, solve for v1 + v2:

v1 + v2 = 48 kg·m/s / 57.0 kg
= 0.8421 m/s (to four decimal places)

Therefore, the speed of Judy and her dog after the catch is approximately 0.8421 m/s.

To find the speed of Judy and her dog after the catch, we can use the principle of conservation of momentum. According to this principle, the total momentum before the catch is equal to the total momentum after the catch.

The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by the formula:

p = m * v

Where:
p = momentum
m = mass
v = velocity

Let's denote the velocity of Judy before the catch as v1, and the velocity of Judy and her dog after the catch as v2.

Before the catch, Judy is stationary, so her initial momentum (p1) is zero, as only the dog is moving. Atti's momentum (p2) before the catch can be calculated using the formula:

p2 = m2 * v2

Where:
m2 = mass of Atti
v2 = velocity of Atti before the catch

After the catch, Judy and Atti move together, so their combined momentum (p3) after the catch can be calculated as:

p3 = (m1 + m2) * v3

Where:
m1 = mass of Judy
m2 = mass of Atti
v3 = velocity of Judy and Atti after the catch

Using the principle of conservation of momentum, we can equate the total momentum before the catch to the total momentum after the catch:

p1 + p2 = p3

Since p1 = 0 (Judy is stationary), the equation simplifies to:

p2 = p3

Now, let's substitute the values into the equation and solve for v3.

p2 = m2 * v2
p3 = (m1 + m2) * v3

m2 * v2 = (m1 + m2) * v3

Plugging in the given values:
m1 = 45.0 kg (mass of Judy)
m2 = 12.0 kg (mass of Atti)
v2 = 4.0 m/s (velocity of Atti before the catch)

(12.0 kg) * (4.0 m/s) = (45.0 kg + 12.0 kg) * v3

48.0 kg·m/s = 57.0 kg * v3

v3 = (48.0 kg·m/s) / (57.0 kg)
v3 ≈ 0.842 m/s

Therefore, the speed of Judy and her dog after the catch is approximately 0.842 m/s.