# Two groups of canoeists meet in the middle of a lake. A person in canoe 1 pushes on canoe 2 with a force of 46 N to separate the canoes. If the mass of canoe 1 and its occupants is 130 kg, and the mass of canoe 2 and its occupants is 250 kg, after 1.20s?

Nov 5, 2015

Canoe 1 : ${V}_{f} = 0.425 \frac{m}{s}$ and $p = 55.2 N \cdot s$

Canoe 2 : ${V}_{f} = 0.221 \frac{m}{s}$ and $p = 55.2 N \cdot s$

#### Explanation:

The crux of this question lies in understanding Newtons 3rd Law:

"Every action will have and equal and opposite reaction"

This means that by pushing with 46N, the canoeist is actually both pushing himself and the other canoe at 46N .

To calculate the final velocity and momentum on Canoe 1 (after 1.20 seconds) you must use Newton's 2nd Law as well, along with the momentum formula.

$F = m \cdot \vec{a}$ and $p = m \cdot {\vec{v}}_{f}$

First use Newton's 2nd law to determine acceleration. $F = m \cdot \vec{a}$

$46 N = 130 K g \cdot \vec{a}$

$\frac{46 N}{130} K g = \vec{a}$

$\vec{a} = 0.354 \frac{m}{s} ^ 2$

Then multiply this acceleration by the time duration given to find Velocity. (one of the seconds from the accleration unit will be canceled leaving you with m/s). This is your final velocity.

$1.2 s \cdot 0.354 \frac{m}{s} ^ 2 = 0.425 \frac{m}{s}$

To find your momentum at that particular time, all you need to do is multiply your newly found final velocity by the mass of the canoe in question (130Kg) using the momentum formula: $p = m \cdot {\vec{v}}_{f}$

$130 K g \cdot 0.425 \frac{m}{s} = 55.2 N \cdot s$

You will use the same steps to find the velocity and momentum of the 2nd canoe, however, you must be sure to use the 2nd canoe's mass. Other than that the process is the same.

If you've done that correctly you will calculate a ${V}_{f} = 0.221 \frac{m}{s}$ and a $p = 55.2 N \cdot s$