A force F applied to an object of mass m causes it to move in a straight line a distance D during an interval of time T. The object gains kinetic energy K during this interval. In which of the following cases will the object gain the same kinetic energy K?
A) A force F is applied to an object of mass 2m during a time interval T
B) A force F is applied to an object of mass 2m as it travels a distance D
C) A force 2F is applied to an object of mass 2m during a time interval T
D) A force 2F is applied to an object of mass 2m as it travels a distance D
To determine which case will result in the same kinetic energy K, we need to understand how kinetic energy is related to force, mass, distance, and time.
The work done on an object is equal to the force applied to the object multiplied by the distance it moves in the direction of the force:
Work = Force x Distance (W = F * D)
The work done on an object is also equal to the change in its kinetic energy:
Work = Change in Kinetic Energy (W = K - 0)
Hence, we have:
K - 0 = F * D
From this, we can conclude that the greater the force applied or the greater the distance moved, the greater the change in kinetic energy.
Now, let's analyze each option to determine if the object will gain the same kinetic energy K:
A) A force F is applied to an object of mass 2m during a time interval T
In this case, only the mass of the object is doubled. The force, distance, and time remain the same. Since the force is the same, but the mass is doubled, the object will not gain the same kinetic energy K.
B) A force F is applied to an object of mass 2m as it travels a distance D
In this case, the mass remains the same, but the distance is doubled. Since the force and mass remain the same, but the distance is doubled, the object will gain the same kinetic energy K. (This is because the work done is directly proportional to the distance moved in the direction of the force.)
C) A force 2F is applied to an object of mass 2m during a time interval T
In this case, the force applied is doubled, but the mass and time remain the same. Since the force and mass are both doubled, the object will gain a greater kinetic energy, not the same as K.
D) A force 2F is applied to an object of mass 2m as it travels a distance D
In this case, the force and mass are both doubled, but the distance remains the same. Since the force and mass are doubled, the object will gain a greater kinetic energy, not the same as K.
Therefore, the only option where the object will gain the same kinetic energy K is option B) A force F is applied to an object of mass 2m as it travels a distance D.
To determine which case will result in the same kinetic energy K, we need to analyze the relationship between force, mass, distance, and time.
First, let's look at the formula for work, which is the transfer of energy from one system to another. In this case, the work done on an object is equal to the change in its kinetic energy:
Work (W) = Change in kinetic energy (ΔK)
Work is calculated by multiplying force (F) by the displacement (D) in the direction of the force:
W = F * D
On the other hand, the formula for kinetic energy is:
Kinetic energy (K) = 0.5 * mass (m) * velocity squared (v²)
Since we know force (F) causes an object to move a certain distance (D) during a given time interval (T), we can rewrite the equation for work as:
W = F * D = ΔK
Now, let's go through each option and see which one results in the same kinetic energy (K):
A) A force F is applied to an object of mass 2m during a time interval T:
In this case, the force is applied for the same time interval (T) but with double the mass (2m). Therefore, the force will create double the change in kinetic energy. Thus, the object will not gain the same kinetic energy.
B) A force F is applied to an object of mass 2m as it travels a distance D:
Here, the force (F) is applied to a mass that is twice the original mass (2m), and it travels the same distance (D). Since the mass is doubled, the resulting kinetic energy will be double as well. Therefore, the object will not gain the same kinetic energy.
C) A force 2F is applied to an object of mass 2m during a time interval T:
In this case, the mass (2m) remains the same, but the force is increased to twice its original value (2F). Since the force is doubled, the resulting change in kinetic energy will also be double. Therefore, the object will not gain the same kinetic energy.
D) A force 2F is applied to an object of mass 2m as it travels a distance D:
In this scenario, both the mass (2m) and the force (2F) are doubled, while the distance (D) remains the same. Doubling both the mass and the force will result in the same change in kinetic energy as before, since they cancel each other out. Therefore, the object will gain the same kinetic energy K.
Therefore, the correct option is D) A force 2F is applied to an object of mass 2m as it travels a distance D.
work done = F D = gain in Ke
Now which of those has the same force times distance (time has nothing to do with it)? We are not doing power, just energy.) The heavier mass will just end up moving slower if they started at rest but will have the same gain in Ke.