Given mass m = 45 kg, initial velocity v = 5 m/s. We assume acceleration due to gravity g = 9.8 m/s2.

Given mass $m = 45 \text{ kg}$, initial velocity $v = 5 \text{ m/s}$. We assume acceleration due to gravity $g = 9.8 \text{ m/s}^2$.
According to the principle of conservation of mechanical energy, the initial kinetic energy is converted into gravitational potential energy at the maximum height.

$$\frac{1}{2}mv^2 = mgh$$

$$h = \frac{(5 \text{ m/s})^2}{2 \times 9.8 \text{ m/s}^2}$$ $$h = \frac{25 \text{ m}^2/\text{s}^2}{19.6 \text{ m/s}^2}$$ $$h \approx 1.28 \text{ m}$$

The height reached is $\boxed{\text{1.28 m}}$.

Question 2:

a) A boy of mass $50 \text{ kg}$ runs and jumps up a small platform with a speed of $6 \text{ m/s}$. How high does he rise?

Given mass $m = 50 \text{ kg}$, initial velocity $v = 6 \text{ m/s}$. We assume acceleration due to gravity $g = 9.8 \text{ m/s}^2$.
Using the principle of conservation of mechanical energy:

$$\frac{1}{2}mv^2 = mgh$$

$$h = \frac{(6 \text{ m/s})^2}{2 \times 9.8 \text{ m/s}^2}$$ $$h = \frac{36 \text{ m}^2/\text{s}^2}{19.6 \text{ m/s}^2}$$ $$h \approx 1.84 \text{ m}$$

The height he rises is $\boxed{\text{1.84 m}}$.

Question 3:

a) A car of mass $800 \text{ kg}$ moves up a slope with a speed of $10 \text{ m/s}$. Find the maximum height it can reach.

Given mass $m = 800 \text{ kg}$, initial velocity $v = 10 \text{ m/s}$. We assume acceleration due to gravity $g = 9.8 \text{ m/s}^2$.
Using the principle of conservation of mechanical energy:

$$\frac{1}{2}mv^2 = mgh$$

$$h = \frac{(10 \text{ m/s})^2}{2 \times 9.8 \text{ m/s}^2}$$ $$h = \frac{100 \text{ m}^2/\text{s}^2}{19.6 \text{ m/s}^2}$$ $$h \approx 5.10 \text{ m}$$

The maximum height the car can reach is $\boxed{\text{5.10 m}}$.

Question 4:

a) A footballer kicks a ball vertically upward with a speed of $12 \text{ m/s}$. Determine the height reached by the ball.

Given initial velocity $u = 12 \text{ m/s}$. We assume acceleration due to gravity $g = 9.8 \text{ m/s}^2$.
Using the principle of conservation of mechanical energy, the initial kinetic energy is converted into gravitational potential energy at the maximum height.

$$\frac{1}{2}mu^2 = mgh$$

$$h = \frac{(12 \text{ m/s})^2}{2 \times 9.8 \text{ m/s}^2}$$ $$h = \frac{144 \text{ m}^2/\text{s}^2}{19.6 \text{ m/s}^2}$$ $$h \approx 7.35 \text{ m}$$

The height reached by the ball is $\boxed{\text{7.35 m}}$.