Which of the following has the maximum momentum

The correct answer is: A 20 g weight after falling 1 kilometre

To determine which object has the maximum momentum, we need to calculate the magnitude of momentum for each given case. The momentum ($p$) of an object is given by the product of its mass ($m$) and its velocity ($v$):

$p=m·v$

Let us calculate the momentum for each option by converting all units into the SI system (kilograms for mass and meters per second for velocity).

Option A: A 100 kg vehicle moving at 0.02 ms^-1
Here, mass $m=100\text{kg}$ and velocity $v=0.02{\text{ms}}^{-1}$.
Calculating momentum:
$p=100\text{kg}\times 0.02{\text{ms}}^{-1}=2{\text{kg ms}}^{-1}$

Option B: A 4 g weight moving at 10000 cms^-1
Converting units to SI:
Mass, $m=4\text{g}=4\times {10}^{-3}\text{kg}=0.004\text{kg}$
Velocity, $v=10000{\text{cms}}^{-1}=100{\text{ms}}^{-1}$
Calculating momentum:
$p=0.004\text{kg}\times 100{\text{ms}}^{-1}=0.4{\text{kg ms}}^{-1}$

Option C: A 200 g weight moving with kinetic energy 10^-6 J
Converting mass to SI:
Mass, $m=200\text{g}=0.2\text{kg}$
Kinetic energy, $K={10}^{-6}\text{J}$
The relation between momentum ($p$), mass ($m$), and kinetic energy ($K$) is:
$p=\sqrt{2mK}$
Calculating momentum:
$p=\sqrt{2\times 0.2\text{kg}\times {10}^{-6}\text{J}}=\sqrt{0.4\times {10}^{-6}}\approx 6.32\times {10}^{-4}{\text{kg ms}}^{-1}$

Option D: A 20 g weight after falling 1 kilometre
Converting units to SI:
Mass, $m=20\text{g}=0.02\text{kg}$
Distance fallen, $h=1\text{km}=1000\text{m}$
Assuming acceleration due to gravity, $g=9.8{\text{ms}}^{-2}$, the velocity ($v$) acquired by a freely falling body is given by:
$v=\sqrt{2gh}=\sqrt{2\times 9.8\times 1000}=\sqrt{19600}=140{\text{ms}}^{-1}$
Calculating momentum:
$p=0.02\text{kg}\times 140{\text{ms}}^{-1}=2.8{\text{kg ms}}^{-1}$

Conclusion:
Comparing the calculated momentum values:
- Option A: $2{\text{kg ms}}^{-1}$
- Option B: $0.4{\text{kg ms}}^{-1}$
- Option C: $6.32\times {10}^{-4}{\text{kg ms}}^{-1}$
- Option D: $2.8{\text{kg ms}}^{-1}$

Thus, the 20 g weight after falling 1 km (Option D) has the maximum momentum.