A rocket motor applies a varying thrust force to a spacecraft over 20 seconds. What is the most direct method to find the spacecraft's change in velocity?
AUse kinematics: find instantaneous acceleration from F = ma at each moment, then integrate twice
BUse work-energy: calculate the work done by thrust over the displacement traveled
CUse impulse-momentum: compute the integral of force over time (area under the F-vs-t curve) and divide by mass
DUse conservation of energy: set the initial kinetic energy equal to the work done by thrust
When force is given as a function of time, impulse-momentum is the natural method: Δv = (∫F dt)/m. The impulse is simply the area under the F-vs-t curve. Work-energy would require knowing force as a function of position — much harder here. The explainer states the method-selection rule directly: 'If the problem gives you a force as a function of time... reach for impulse-momentum.' Time as the key variable → impulse-momentum; displacement as the key variable → work-energy.
Question 2 Multiple Choice
A bat strikes a baseball over a contact time of 0.002 s, changing its momentum by 15 kg·m/s. An observer then argues 'momentum was conserved because the collision was brief.' What key condition is actually required for momentum conservation?
AThe collision must be perfectly elastic — kinetic energy must also be conserved
BNet external impulse must be zero in the direction of interest during the time interval
CBoth objects must have the same mass for momentum exchange to balance
DOne object must be stationary before the collision
Momentum conservation requires that net external impulse (∫F_external dt) be zero in the relevant direction. During the bat-ball collision, the bat and ball exert equal and opposite internal forces on each other — but the bat is swung by a player and connected to the ground through arms and legs, providing substantial external impulse. The system's momentum is not conserved. The collision being brief reduces gravitational impulse but doesn't eliminate the bat's external driving force. 'Brief' and 'conserved' are not synonyms.
Question 3 True / False
The impulse-momentum principle applies independently in each coordinate direction, so linear momentum can be conserved in the x-direction while not conserved in the y-direction.
TTrue
FFalse
Answer: True
Momentum is a vector, and the impulse-momentum principle gives separate equations per direction: ∫F_x dt = Δ(mv_x) and ∫F_y dt = Δ(mv_y). If there is no external horizontal impulse (no friction, no horizontal applied force), x-momentum is conserved. If gravity acts vertically throughout the interaction, it contributes an impulse mg·Δt in the y-direction, so y-momentum is not conserved. The explainer states: 'conserved in x does not imply conserved in y.'
Question 4 True / False
In impulse-momentum analysis, the impulse due to gravity can generally be neglected because collisions happen too quickly for gravity to have any effect.
TTrue
FFalse
Answer: False
Whether gravitational impulse is negligible depends entirely on the duration of the interaction. For very brief impacts — a bat-ball contact of ~0.001 s — the gravitational impulse mg·Δt is tiny compared to the contact impulse and can be neglected. But for slower interactions lasting seconds — a rocket firing, two skaters pushing apart, a ball rolling — mg·Δt accumulates to a significant value. The explainer distinguishes these cases explicitly: gravitational impulse 'is often small for very brief impacts... but is significant for slower interactions.'
Question 5 Short Answer
Why is the impulse-momentum method preferred over directly applying Newton's second law (F = ma) when a force varies with time?
Think about your answer, then reveal below.
Model answer: Impulse-momentum directly integrates force over time to get momentum change in one step: ΔL = ∫F dt, so Δv = ΔL/m. Applying F = ma to a time-varying force would require computing acceleration as a function of time and then integrating again to get velocity — two integration steps instead of one. Impulse-momentum matches the problem structure: when the input is a force-vs-time relationship and the output is a velocity change, the method's variables align perfectly with the given information.
Both methods are mathematically equivalent — impulse-momentum is simply Newton's second law integrated once over time. The practical advantage is that impulse-momentum bypasses acceleration entirely. This is especially powerful for impact problems where average force is measured from a known velocity change and a measured contact time: F_avg = ΔL/Δt requires no trajectory information at all.