Your physical therapist throws a baseball to you at a certain speed and you catch it. To increase the difficulty, the therapist is going to throw you a medicine ball whose mass is ten times the mass of the baseball. You are given the following choices. Rank these from easiest to hardest to catch.
1. The medicine ball thrown at the same speed as the baseball
2. The medicine ball thrown with the same momentum as the baseball.
3. The medicine ball thrown with the same kinetic energy as the baseball.
The sequence would be 2,3,1. Looking at a specific variable, lets say the velocity of the larger mass.
For the first kind of catch, the velocity of the medicine ball is the same, so it should be the hardest as there is no compensation for difficulty due to the increase in mass.
The second kind of catch, is the easiest as the relationship between momentum is (m1)*(v1)=(m2)*(v2).
Since we know that the mass of the medicine ball is ten time the mass of the baseball, this can be rewritten as v2=(1/10)v1. (The 1/10 is from the mass relations, and the mass values cancel out (m1/10m1).
The third kind of catch is a little bit trickier to calculate. By substituting 10(m1) in for (m2) though, it can be derived. The equation for kinetic energy is KE=.5mv^2. If the KE is equal, then the statement (m1)(v1)^2=(m2)(v2)^2 should be true. Since 10m1=m2, it can be further derived that .1(v1)^2=(v2)^2. So V2=sqrt(.1*(v1)^2). This is greater than the second method(since sqrt(.1)>.1), but less than the first. So the order is 2, 3 then finally 1.
Think about that answer. If you catch something 10 times heavier going at the same speed, it's kinetic energy and its momentum are both going to be multiplied by 10. So given that you have a choice between the same momentum, the same kinetic energy, or 10 times as much momentum and kinetic energy, I'd say the choice "a" is by far the worst option. In order to keep the same momentum and kinetic energy for the heavier ball, it's speed would have to be reduced, so b and c would make much more sense.
To rank these choices from easiest to hardest to catch, we need to consider the factors involved in catching an object.
1. The medicine ball thrown at the same speed as the baseball:
If the medicine ball is thrown at the same speed as the baseball, it means they have the same velocity. Since the velocity is the same, catching the medicine ball would not be significantly harder than catching the baseball. The mass difference does not affect the difficulty in this scenario. Therefore, this option would be the easiest to catch.
2. The medicine ball thrown with the same momentum as the baseball:
Momentum is the product of an object's mass and velocity. So if the medicine ball has ten times the mass of the baseball but the same momentum, it means that the medicine ball would be thrown at a lower velocity than the baseball. Catching a slower-moving medicine ball would be easier than catching the baseball thrown at a higher speed. Hence, this option would be harder to catch compared to the first one.
3. The medicine ball thrown with the same kinetic energy as the baseball:
Kinetic energy is the energy possessed by an object due to its motion. It depends on both the mass and velocity of the object. If the medicine ball has ten times the mass of the baseball but the same kinetic energy, it would need to be thrown at a much higher velocity. Catching an object moving at a higher velocity requires more force and precision, making it more challenging to catch. Therefore, this option would be the hardest to catch out of the three choices.
So the ranking from easiest to hardest to catch is:
1. The medicine ball thrown at the same speed as the baseball.
2. The medicine ball thrown with the same momentum as the baseball.
3. The medicine ball thrown with the same kinetic energy as the baseball.