1.4.2 · D5Momentum & Collisions
Question bank — Impulse-momentum theorem — derivation
Before we start, three words must already be clear — if any feels shaky, re-read the parent first:
True or false — justify
Every answer below explains why, because a bare "true/false" hides the trap.
True or false: A small force can produce the same momentum change as a large force.
True — if the small force acts long enough. Impulse is , so a small paired with a big gives the same .
True or false: Impulse and momentum have different units.
False — both are . Since and equal things must share units, is .
True or false: If the net impulse on an object is zero, its velocity is unchanged.
True — zero net impulse means , so ; with fixed mass, .
True or false: Impulse is always positive because it comes from a force.
False — impulse is a vector and carries a sign. A force opposing motion (like a catching hand) gives a negative impulse in the direction of travel.
True or false: A ball that bounces back experiences a larger impulse than one that just stops, given the same incoming speed.
True — stopping gives ; bouncing gives because the velocity reverses, so the impulse doubles.
True or false: is the fundamental definition of impulse.
False — the fundamental definition is . The form is only a special case valid when is constant.
True or false: The impulse–momentum theorem only works for constant forces.
False — it is derived by integrating , so it holds for any force, however wild, as long as you use the integral (area under the curve).
True or false: A heavy object and a light object hit by the same impulse gain the same velocity.
False — they gain the same momentum change , but , so the lighter object speeds up more.
Spot the error
Each statement contains one flaw. Find it and repair it.
"A ball hits a wall at and bounces at , so its momentum is unchanged."
The speed is unchanged but the velocity reversed (), so . Momentum is a vector; direction counts.
"The force during a collision is , so the impulse is ."
Impulse is force times time, — you cannot get from alone. Multiply by the contact duration.
"Since , we can write even when the force spikes."
If the force spikes, is not constant, so uses a single acceleration that never existed. Use = area under the curve instead.
"Impulse = area under the velocity–time graph."
Wrong graph. Impulse is the area under the force–time graph. Area under a velocity–time graph is displacement.
"An airbag reduces the impulse on the driver during a crash."
The impulse is nearly fixed — the driver's momentum must still drop to zero. The airbag increases , which reduces the average force , not the impulse.
"A force acts for giving impulse , so the peak force was ."
That gives the average force (), not the peak. A spiky force has a peak higher than its average while enclosing the same area.
"Catching an egg gently changes the impulse, keeping it whole."
The impulse (needed to stop the egg) is the same either way. Moving your hands back stretches , lowering the force below the egg's breaking point.
Why questions
Why does the momentum form appear in the derivation instead of ?
Because secretly assumes constant mass; is the more fundamental form and integrates cleanly to .
Why do we multiply Newton's law by before integrating?
To gather the total effect over a time interval — turns "rate of change now" into "accumulated push," ready to sum from to .
Why does bending your knees when you land soften the impact?
Your momentum change to zero is fixed, but bending stretches the stopping time , so drops.
Why is impulse useful when we don't know the detailed force during a collision?
Because lets us find the net effect from measurable start and end velocities, without ever knowing the messy .
Why must we assign a direction before solving any impulse problem?
Impulse and momentum are vectors; a consistent sign convention keeps a "reversal" (doubling) from accidentally looking like a "cancellation."
Why is impulse called the "time-analogue" of work?
Work is force accumulated over distance () and changes energy; impulse is force accumulated over time () and changes momentum — see Work–Energy Theorem.
Edge cases
What is the impulse on an object during a time interval in which the net force is exactly zero?
Zero — no force means no accumulated push, so and momentum is conserved over that interval.
If an object's speed is constant but its direction changes (circular motion), is the net impulse zero?
No — velocity is a vector, so a direction change means , hence and a real net impulse acted.
An infinitely hard collision would give an instantaneous () stop. What happens to the average force?
— with fixed and shrinking to zero, the force blows up. This is why rigid impacts are so damaging.
A force acts, then reverses and acts equally the other way over the interval. What is the net impulse?
Zero — the positive and negative areas under the force–time graph cancel, so even though a force was present the whole time.
If two objects exert forces on each other (a collision), how do their impulses compare?
They are equal and opposite — by Newton's third law the forces are equal and opposite at every instant, so their time-integrals (impulses) are too. This is the seed of Conservation of Momentum.
For a ball dropped and caught at the same speed it left (perfect bounce), is the impulse from the floor zero?
No — the velocity reverses, so ; the floor must supply an impulse of upward to flip the motion.
Connections
- Impulse-momentum theorem — derivation — the parent derivation these traps test.
- Newton's Second Law — the momentum form that seeds the whole theorem.
- Conservation of Momentum — the zero-net-impulse and equal-and-opposite cases above.
- Force–Time Graphs — "impulse = area" traps live here.
- Work–Energy Theorem — the distance-analogue contrasted in the "why" section.
- Collisions — Elastic and Inelastic — bounce-vs-stick impulse doubling in action.