Matt 86.0 kg and his dog 18.0 kg are playing around on a frozen lake. Matt attempts to run across the ice at 2.90 m/s and his dog runs toward him at 1.40 m/s. If his dog jumps up into his arms, what is Matt's speed just after he catches his dog?
86 * 2.9 - 18 * 1.4 = (86+18)v
To solve this problem, we need to use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after an event.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is calculated by multiplying the mass (m) of an object by its velocity (v).
The initial momentum of Matt can be calculated by multiplying his mass (86.0 kg) by his initial velocity (2.90 m/s). So, his initial momentum is:
Initial momentum of Matt = mass of Matt × initial velocity of Matt
= 86.0 kg × 2.90 m/s
= 249.4 kg·m/s
Similarly, the initial momentum of the dog can be calculated by multiplying its mass (18.0 kg) by its initial velocity (1.40 m/s). So, the initial momentum of the dog is:
Initial momentum of the dog = mass of the dog × initial velocity of the dog
= 18.0 kg × 1.40 m/s
= 25.2 kg·m/s
Since this is an isolated system and no external forces are acting on Matt and his dog, the total momentum before the dog jumps will be the sum of their individual momenta. Therefore, the total initial momentum of the system before the dog jumps is:
Total initial momentum = Initial momentum of Matt + Initial momentum of the dog
= 249.4 kg·m/s + 25.2 kg·m/s
= 274.6 kg·m/s
Now, when the dog jumps up and Matt catches him, their combined system will have a final momentum. Since they both move together after the jump, their combined final momentum will be the same as the initial momentum of the system before the jump.
Final momentum = Total initial momentum
= 274.6 kg·m/s
Since momentum is the product of mass and velocity, we can use this equation to calculate Matt's velocity (v) just after he catches his dog:
Final momentum (which is equal to the momentum of Matt and the dog) = total mass of Matt and the dog × final velocity
= (mass of Matt + mass of the dog) × final velocity
Rearranging the equation, we can solve for the final velocity:
Final velocity = Final momentum / (mass of Matt + mass of the dog)
Substituting the known values into the equation, we can find the final velocity of Matt:
Final velocity = 274.6 kg·m/s / (86.0 kg + 18.0 kg)
= 274.6 kg·m/s / 104.0 kg
≈ 2.64 m/s
Therefore, Matt's speed just after he catches his dog is approximately 2.64 m/s.