GRE Physics Companion: Collisions in One Dimension

GRE Physics Companion: Collisions in One Dimension

For every isolated one-dimensional collision,

(1)

The second equation depends on collision type:

elastic:	Ki = Kf,	(2)
perfectly inelastic:	v1 = v2,	(3)
general restitution:	v2 − v1 = e(u1 − u2).	(4)

1 High-value GRE facts

1. Momentum is conserved when external impulse is negligible.
2. kinetic energy is conserved only in elastic collisions.
3. Perfectly inelastic means the objects stick together.
4. Equal masses exchange velocities in a one-dimensional elastic collision.
5. For an elastic collision, separation speed equals approach speed.
6. For a target initially at rest, use the standard elastic formulas only if the collision is elastic.
7. e = 1 is elastic and e = 0 is perfectly inelastic.
8. In the center of mass frame, elastic collision velocities reverse.
9. Equal-and-opposite collision impulses do not imply equal velocity changes for unequal masses.
10. Always keep the velocity signs.

Part I: Original GRE-style problems

Problem 1: perfectly inelastic velocity

A mass m moving at speed v sticks to an identical stationary mass. Their final speed is

1. 0
2. v∕4
3. v∕2
4. v
5. 2v

Problem 2: perfectly inelastic kinetic energy

For the collision in Problem 1, the final translational kinetic energy is what fraction of the initial kinetic energy?

1. 1∕4
2. 1∕2
3. 3∕4
4. 1
5. 2

Problem 3: equal-mass elastic collision

A mass m moving at +v collides elastically with an identical stationary mass. The first mass leaves with velocity

1. −v
2. −v∕2
3. 0
4. v∕2
5. v

Problem 4: elastic relative speed

For a one-dimensional elastic collision,

1. v₁ + v₂ = u₁ + u₂ always
2. v₂ − v₁ = u₁ − u₂
3. v₁ − v₂ = u₁ − u₂
4. v₁ + v₂ = 0
5. both final velocities must be positive

Problem 5: restitution

A collision has coefficient of restitution e = 0.40 and approach relative speed 10 m∕s. The separation relative speed is

1. 2 m∕s
2. 4 m∕s
3. 6 m∕s
4. 10 m∕s
5. 25 m∕s

Problem 6: heavy wall limit

A Light ball collides elastically with a stationary wall of effectively infinite mass. The ball’s final velocity is approximately

1. 0
2. +u∕2
3. +u
4. −u∕2
5. −u

Problem 7: heavy projectile limit

A very heavy object moving at speed u strikes a very light stationary object elastically. The light object’s final speed approaches

1. 0
2. u∕2
3. u
4. 2u
5. 4u

Problem 8: kinetic energy and restitution

For fixed masses and initial relative speed, the fraction of relative kinetic energy remaining after impact is

1. e
2. e²
3. 1 − e
4. 1 − e²
5. 2e

Problem 9: collision impulse

During a collision, object 1 receives impulse −7 N s. External impulse is negligible. Object 2 receives

1. −14 N s
2. −7 N s
3. 0
4. +7 N s
5. +14 N s

Problem 10: perfectly inelastic center of mass

Two objects stick together in an isolated collision. Their common final velocity equals

1. the faster initial velocity
2. the slower initial velocity
3. the initial center of mass velocity
4. zero in every case
5. the arithmetic average of the initial speeds

Problem 11: target at rest, elastic

A mass m moving at speed u elastically strikes a stationary mass 3m. The first mass’s final velocity is

1. −u
2. −u∕2
3. 0
4. u∕2
5. u

Problem 12: identifying collision type

An isolated collision conserves momentum and leaves the two bodies moving together afterward. The collision is

1. necessarily elastic
2. perfectly inelastic
3. impossible
4. superelastic
5. a non-collision

Part II: Complete worked solutions

Solution 1

Momentum conservation gives

(5)

Thus

(6)

Answer: (C).

Solution 2

Initially,

(7)

Finally,

Kf	= (2m)2	(8)
	= mv2.	(9)

Therefore

(10)

Answer: (B).

Solution 3

Equal masses exchange velocities in a one-dimensional elastic collision.

The moving mass stops:

(11)

Answer: (C).

Solution 4

For elastic one-dimensional impact,

(12)

Answer: (B).

Solution 5

By definition,

(13)

Thus

(14)

Answer: (B).

Solution 6

For m₂ ≫ m₁ and u₂ = 0,

(15)

Answer: (E).

Solution 7

For m₁ ≫ m₂,

(16)

Answer: (D).

Solution 8

Because relative speed is multiplied by e and kinetic energy depends on speed squared,

(17)

Answer: (B).

Solution 9

Internal collision impulses are equal and opposite:

(18)

Thus

(19)

Answer: (D).

Solution 10

When the objects stick, their common final velocity is

(20)

Answer: (C).

Solution 11

For an elastic collision with the target at rest,

(21)

With m₁ = m and m₂ = 3m,

(22)

Answer: (B).

Solution 12

If two bodies stick together after impact, the collision is perfectly inelastic.

Answer: (B).

2 GRE checklist

1. Choose a positive direction and keep every velocity sign.
2. Write momentum conservation first.
3. Identify the collision type before choosing the second equation.
4. Elastic means conserve kinetic energy.
5. Perfectly inelastic means set v₁ = v₂.
6. For restitution problems, use v₂ − v₁ = e(u₁ − u₂).
7. Check equal-mass and heavy-mass limiting cases.
8. Do not assume equal-and-opposite impulses produce equal speed changes.

References

References