A 2000 kg car moving at 5 m/s collides inelastically with a 5000 kg minibus moving at -7 m/s. Find the final velocity and the impulse on the minibus.

Since the vehicles stick together, it's an inelastic collision, and they will have a common final velocity, $V$.

$$m_c u_c + m_m u_m = (m_c + m_m) V$$

$$(2000\, \text{kg})(5\, \text{m/s}) + (5000\, \text{kg})(-7\, \text{m/s}) = (2000\, \text{kg} + 5000\, \text{kg}) V$$ $$10000\, \text{kg}\cdot\text{m/s} - 35000\, \text{kg}\cdot\text{m/s} = (7000\, \text{kg}) V$$ $$-25000\, \text{kg}\cdot\text{m/s} = (7000\, \text{kg}) V$$

$$V = \frac{-25000\, \text{kg}\cdot\text{m/s}}{7000\, \text{kg}}$$ $$V = -3.571428... \, \text{m/s}$$

Rounding to 3 decimal places:

$$V = -3.571\, \text{m/s}$$

The negative sign indicates that the combined system moves in the initial direction of the minibus.
The velocity of the system after collision is $\boxed{\text{-3.571 m/s}}$.

b) Calculate the impulsive force on the minibus.

Impulsive force is the rate of change of momentum.
Impulse, $J = \Delta p = m_m V - m_m u_m$

$$J = (5000\, \text{kg})(-3.571428\, \text{m/s}) - (5000\, \text{kg})(-7\, \text{m/s})$$ $$J = -17857.14\, \text{kg}\cdot\text{m/s} + 35000\, \text{kg}\cdot\text{m/s}$$ $$J = 17142.86\, \text{N}\cdot\text{s}$$

$$F = \frac{17142.86\, \text{N}\cdot\text{s}}{0.2\, \text{s}}$$ $$F = 85714.3\, \text{N}$$

The impulsive force on the minibus is $\boxed{\text{85714.3 N}}$.

c) Calculate the change in kinetic energy of the system to 5 significant figures.

$$KE_{initial} = \frac{1}{2} m_c u_c^2 + \frac{1}{2} m_m u_m^2$$ $$KE_{initial} = \frac{1}{2} (2000\, \text{kg}) (5\, \text{m/s})^2 + \frac{1}{2} (5000\, \text{kg}) (-7\, \text{m/s})^2$$ $$KE_{initial} = \frac{1}{2} (2000\, \text{kg}) (25\, \text{m}^2/\text{s}^2) + \frac{1}{2} (5000\, \text{kg}) (49\, \text{m}^2/\text{s}^2)$$ $$KE_{initial} = 25000\, \text{J} + 122500\, \text{J}$$ $$KE_{initial} = 147500\, \text{J}$$

$$KE_{final} = \frac{1}{2} (m_c + m_m) V^2$$ $$KE_{final} = \frac{1}{2} (2000\, \text{kg} + 5000\, \text{kg}) (-3.571428\, \text{m/s})^2$$ $$KE_{final} = \frac{1}{2} (7000\, \text{kg}) (12.755102\, \text{m}^2/\text{s}^2)$$ $$KE_{final} = 3500\, \text{kg} \times 12.755102\, \text{m}^2/\text{s}^2$$ $$KE_{final} = 44642.857\, \text{J}$$

$$\Delta KE = 44642.857\, \text{J} - 147500\, \text{J}$$ $$\Delta KE = -102857.143\, \text{J}$$

Rounding to 5 significant figures:

$$\Delta KE = -102860\, \text{J}$$

The change in kinetic energy of the system is $\boxed{\text{-102860 J}}$.

Question 18:

a) (i) State the law of floatation.

The law of floatation states that an object floats if the upthrust acting on it is equal to its weight. This occurs when the object displaces a volume of fluid whose weight is equal to the object's weight.

a) (ii) Explain why a hollow metal sphere floats on water while a solid metal sphere of the same material sinks in water.

A hollow metal sphere floats because its overall average density (total mass divided by total volume, including the air inside) is less than the density of water. A solid metal sphere of the same material sinks because its density is greater than the density of water. The hollow sphere displaces enough water to generate an upthrust equal to its weight, while the solid sphere does not.