Question: Finding the mass of a Cruise Ship
A tugboat pulls a cruise ship out of port. You estimate the acceleration by noting that the tugboat takes 60s to move the cruise ship 100m, starting from rest. If the tugboat exerts 3x10^6N of thrust, what is the mass of the cruise ship? Ignore drag due to the water, and assume that the tugboat accelerates uniformly.
1. **State the problem:**
We need to find the mass $m$ of the cruise ship given the force $F = 3 \times 10^{6}$ N, the displacement $s = 100$ m, and the time $t = 60$ s, starting from rest with uniform acceleration.
2. **Relevant formulas:**
Newton's second law: $$F = m a$$
Kinematic equation for displacement with uniform acceleration starting from rest: $$s = \frac{1}{2} a t^{2}$$
3. **Find acceleration $a$ using the kinematic equation:**
$$s = \frac{1}{2} a t^{2} \implies a = \frac{2s}{t^{2}}$$
Substitute values:
$$a = \frac{2 \times 100}{60^{2}} = \frac{200}{3600} = \frac{200}{3600}$$
Simplify fraction:
$$a = \frac{\cancel{200}}{\cancel{3600}} = \frac{5}{90} = \frac{1}{18} \approx 0.0556 \text{ m/s}^2$$
4. **Use Newton's second law to find mass $m$:**
$$F = m a \implies m = \frac{F}{a}$$
Substitute values:
$$m = \frac{3 \times 10^{6}}{\frac{1}{18}} = 3 \times 10^{6} \times 18 = 5.4 \times 10^{7} \text{ kg}$$
5. **Final answer:**
The mass of the cruise ship is approximately $$5.4 \times 10^{7}$$ kilograms.