A constant force acts on two different objects that are initially at rest and have different masses, but the force is applied for the same duration of time on each. Which physical quantity will be the same for both objects after the force has acted?

Introduction / Context:
This question combines basic ideas from Newton laws of motion with the concept of momentum. It asks you to reason about what happens when the same force is applied for the same time to objects of different masses that start from rest. Understanding the relationship between force, time, and change in momentum is crucial for many problems in mechanics and collisions.

Given Data / Assumptions:

• Two objects have different masses.
• Both objects are initially at rest, so their initial velocities are zero.
• The same constant force F acts on each object.
• The force is applied for the same time interval t for both objects.
• We are asked which physical quantity is the same for both after the interaction.

Concept / Approach:
Newton second law in impulse form states that the change in momentum (Δp) of an object is equal to the impulse applied to it: Δp = F * t If the same force F acts for the same time t on both objects, then F * t is the same for both, so the change in momentum is the same. Since both start from rest, their initial momentum is zero, which means their final momentum magnitudes are equal. However, because their masses differ, their accelerations and final velocities will be different, and thus their kinetic energies will also differ.

Step-by-Step Solution:
Step 1: Use the impulse–momentum relation: Δp = F * t. Step 2: For each object, the same force F acts for the same time t, so the impulse F * t is identical. Step 3: Since both objects start from rest, their initial momentum is zero. Step 4: Therefore, final momentum p_final = Δp = F * t is the same for both objects. Step 5: Conclude that momentum is the physical quantity that will be equal for both objects after the force acts.

Verification / Alternative check:
Let object 1 have mass m1 and object 2 have mass m2. After time t under the same force F, accelerations are a1 = F / m1 and a2 = F / m2, which are different if the masses differ. The velocities after time t are v1 = a1 * t and v2 = a2 * t. However, the momentum p1 = m1 * v1 = m1 * (F / m1) * t = F * t, and p2 = m2 * v2 = m2 * (F / m2) * t = F * t. Thus, p1 and p2 are equal, confirming that the momentum is the same.

Why Other Options Are Wrong:

• Acceleration: Since acceleration a = F / m, different masses experience different accelerations under the same force, so acceleration is not equal.
• Kinetic energy: Kinetic energy depends on mass and velocity (KE = 0.5 * m * v^2). Because masses and velocities differ, kinetic energies will generally be different.
• Velocity: Objects with different masses under the same force will have different accelerations and therefore different velocities after the same time interval.

Common Pitfalls:
Students sometimes assume that because the same force and time are used, velocities must be the same, overlooking how acceleration depends on mass. Others think kinetic energy, rather than momentum, is equal because the same work might be assumed to be done, which is not true in this setup. Remember that impulse directly relates to change in momentum, not to velocity or kinetic energy alone.

Final Answer:
The physical quantity that will be the same for both objects is momentum.