A baseball has a mass of 135 grams and a softball has a mass of 270 grams. In which of the following situations would they have the same momentum?
The baseball and softball are thrown at the same speed in opposite directions.
The softball is thrown twice as fast as the baseball
in opposite directions.
The baseball is thrown twice as fast as the softball
in the same direction.
The baseball and softball are thrown at the same speed in the same direction.
To determine the situation in which the baseball and softball have the same momentum, we need to consider the momentum formula, which is given by:
momentum = mass × velocity
Let's evaluate the momentum for each situation:
1. The baseball and softball are thrown at the same speed in opposite directions:
Since the speed is the same for both objects, only the direction is different. Therefore, the velocities have opposite signs. Let's denote the magnitude of the speed by v.
For the baseball:
momentum_baseball = mass_baseball × velocity_baseball
momentum_baseball = 135 g × (-v)
For the softball:
momentum_softball = mass_softball × velocity_softball
momentum_softball = 270 g × v
The magnitudes should be equal, so we have:
135 g × (-v) = 270 g × v
-135v = 270v
-135 = 270
The equation has no solutions; thus, the baseball and softball will not have the same momentum in this situation.
2. The softball is thrown twice as fast as the baseball in opposite directions:
Denoting the magnitude of the speed of the baseball by v_baseball and the magnitude of the speed of the softball by v_softball (where v_softball = 2v_baseball), we can use the momentum formula:
For the baseball:
momentum_baseball = mass_baseball × velocity_baseball
momentum_baseball = 135 g × v_baseball
For the softball:
momentum_softball = mass_softball × velocity_softball
momentum_softball = 270 g × (-v_softball)
momentum_softball = 270 g × (-2v_baseball)
momentum_softball = -540 g × v_baseball
To find when the momentum is the same, we can set up the equation:
135 g × v_baseball = -540 g × v_baseball
Dividing both sides by v_baseball (which is not 0), we get:
135 g = -540 g
This equation has no solutions, meaning that the baseball and softball will not have the same momentum in this situation.
3. The baseball is thrown twice as fast as the softball in the same direction:
Similar to the previous situation, let's denote the magnitude of the speeds as v_baseball and v_softball (where v_baseball = 2v_softball):
For the baseball:
momentum_baseball = mass_baseball × velocity_baseball
momentum_baseball = 135 g × v_baseball
For the softball:
momentum_softball = mass_softball × velocity_softball
momentum_softball = 270 g × v_softball
To find when the momentum is the same, we set up the equation:
135 g × v_baseball = 270 g × v_softball
Since v_baseball = 2v_softball, we can substitute:
135 g × (2v_softball) = 270 g × v_softball
270 g × v_softball = 270 g × v_softball
The equation is true for any value of v_softball, indicating that the baseball and softball will have the same momentum in this situation.
4. The baseball and softball are thrown at the same speed in the same direction:
For the baseball:
momentum_baseball = mass_baseball × velocity_baseball
momentum_baseball = 135 g × v
For the softball:
momentum_softball = mass_softball × velocity_softball
momentum_softball = 270 g × v
The magnitudes are equal, so the equation is:
135 g × v = 270 g × v
We can cancel out v (which is not zero) to get:
135 g = 270 g
This equation is not true, so the baseball and softball will not have the same momentum in this situation.
In conclusion, the situation in which the baseball and softball have the same momentum is when the baseball is thrown twice as fast as the softball in the same direction.