Final answer:
The ratio of momentum A to momentum B is 4, because momentum is directly proportional to mass. The ratio of kinetic energy A to kinetic energy B is also 4 since kinetic energy is proportional to the mass and the velocity squared, and the velocities are identical. Both ratios are the same due to the identical velocities of the objects.
Step-by-step explanation:
When two objects, A and B, have identical velocities but object A has 4 times the mass of object B, we can find the momentum and kinetic energy ratios as follows:
- Momentum is the product of an object's mass and its velocity (p = mv). Therefore, the momentum of object A (pA) is 4 times greater than the momentum of object B (pB) since they have the same velocity but object A has 4 times the mass. The ratio of momentum A to momentum B is pA/pB = (4m*v)/(m*v) = 4.
- Since kinetic energy is given by the formula KE = 1/2 * m * v^2, object A's kinetic energy (KEA) will be 4 times greater as well due to its mass being 4 times greater. However, because kinetic energy depends on the square of the velocity, and the velocities are the same, the ratio of kinetic energy A to kinetic energy B is still KEA/KEB = (1/2 * 4m * v^2) / (1/2 * m * v^2) = 4.
In both cases, the ratios are the same because velocity is identical for both objects; however, the mass influences both momentum and kinetic energy linearly in the case of momentum and quadratically in the case of kinetic energy.