A roller-coaster car of mass 200 kg, with the engine switched off, travels along track ABC which has a rough surface, as shown in the diagram below - NSC Physical Sciences - Question 5 - 2020 - Paper 1

The change in kinetic energy (ΔK) can be calculated using the formula:

egin{align*} ext{Initial Kinetic Energy (K}_A ext{)} &= rac{1}{2} m v_A^2 \\ ext{Final Kinetic Energy (K}_B ext{)} &= rac{1}{2} m v_B^2 \\ ext{ΔK} &= K_B - K_A egin{align*}

Substituting the given values:

K_A = rac{1}{2} imes 200 imes (4)^2 = 1600 ext{ J} \\ K_B = rac{1}{2} imes 200 imes (2)^2 = 400 ext{ J}

Calculating the change:

$$ΔK = 400 - 1600 = -1200 ext{ J}$$

Applying the principle of conservation of energy, the total mechanical energy at point A equals the total mechanical energy at point B plus the work done against friction:

$$mg(h + 10) + K_A - W = mg(h) + K_B$$

Substituting the known values, where m = 200 kg:

$$200 imes 9.8 imes 10 + 1600 - 3.40 imes 10^3 = 200 imes 9.8 imes h + 400$$

This simplifies to:

$$19600 + 1600 - 3400 = 1960h + 400$$

Further simplifying gives:

$$19600 + 1600 - 3400 - 400 = 1960h$$ $$= 19600 - 3400 - 400 = 1960 h$$

The final rearrangement yields:

h = rac{19600 - 3400 - 400}{1960} \\ = rac{ 17800}{1960} \\ ≈ 9.08 ext{ m}

Power can be calculated using the formula:

P = rac{W}{t}

Where W is the work done. The total work done against friction during the movement from point B to point C can be calculated using:

$$W = F imes d$$

The distance (d) the car travels from B to C is:

$$d = v imes t = 2 imes 5 = 10 ext{ m}$$

The work done against friction:

$$W_{friction} = 50 imes 10 = 500 ext{ J}$$

Thus, the total work done by the engine, considering the change in gravitational potential energy moving up (ΔPE), is:

$$W = W_{friction} + ΔPE = 500 + (200)(9.8)(22 - h)$$

Calculating ΔPE gives:

$$ΔPE = 200 imes 9.8 imes (22 - 10)$$

Calculating the total:

$$W = 500 + (200)(9.8)(12) \\ = 500 + 23520 \\ = 24020 ext{ J}$$

Finally, substituting into the power formula:

P = rac{24020}{5} = 4804 ext{ W}