1 Calculate h, the change in its vertical height. 4.2 Analyse the equation and calculation that you used to answer Question 4.1 and explain whether you needed to know the mass of the trolley to do the calculation.

Question

Asked on August 10, 2026Physics

This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

Accepted Answer · 2026-08-10T17:30:51.598Z

Here are the calculations for the questions: Question 4: A trolley of mass 1 kg rolls down an inclined plane. The trolley on the inclined plane has an initial velocity of 0.2 m/s. When it reaches the bottom its velocity is 2.2 m/s. 4.1 Calculate h, the change in its vertical height. Step 1: Apply the principle of conservation of mechanical energy. Let the initial height be h and the final height be 0 m. (1)/(2)mv_i^2 + mgh_i = (1)/(2)mv_f^2 + mgh_f Step 2: Substitute the given values (m = 1 kg, v_i = 0.2 m/s, h_i = h, v_f = 2.2 m/s, h_f = 0 m, g = 9.8 m/s^2). (1)/(2)(1 kg)(0.2 m/s)^2 + (1 kg)(9.8 m/s^2)h = (1)/(2)(1 kg)(2.2 m/s)^2 + (1 kg)(9.8 m/s^2)(0 m) Step 3: Simplify and solve for h. 0.5(0.04) + 9.8h = 0.5(4.84) + 0 0.02 + 9.8h = 2.42 9.8h = 2.42 - 0.02 9.8h = 2.40 h = (2.40)/(9.8) h ≈ 0.24 m The change in vertical height is 0.24 m. 4.2 Analyse the equation and calculation that you used to answer Question 4.1 and explain whether you needed to know the mass of the trolley to do the calculation. The equation used is the conservation of mechanical energy: (1)/(2)mv_i^2 + mgh_i = (1)/(2)mv_f^2 + mgh_f. If we divide every term by the mass m, the equation becomes (1)/(2)v_i^2 + gh_i = (1)/(2)v_f^2 + gh_f. This shows that the mass m cancels out. Therefore, you did not need to know the mass of the trolley to calculate the change in vertical height. 4.3 Given the same situation and the same initial and final velocities, would the change in vertical height be the same for a 2 kg trolley? Yes, the change in vertical height would be the same for a 2 kg trolley. As explained in 4.2, the mass cancels out in the conservation of mechanical energy equation when calculating height change based on velocity changes. Question 5: Figure 8 shows a pendulum bob of mass 100 g that is 100 mm above its lowest point and has a speed of 2 m/s. Calculate its speed when it is 50 mm above its lowest point. Step 1: Convert units to SI units. m = 100 g = 0.1 kg h_1 = 100 mm = 0.1 m h_2 = 50 mm = 0.05 m Step 2: Apply the principle of conservation of mechanical energy. (1)/(2)mv_1^2 + mgh_1 = (1)/(2)mv_2^2 + mgh_2 Step 3: Substitute the known values (m = 0.1 kg, v_1 = 2 m/s, h_1 = 0.1 m, h_2 = 0.05 m, g = 9.8 m/s^2). (1)/(2)(0.1 kg)(2 m/s)^2 + (0.1 kg)(9.8 m/s^2)(0.1 m) = (1)/(2)(0.1 kg)v_2^2 + (0.1 kg)(9.8 m/s^2)(0.05 m) Step 4: Simplify and solve for v_2^2. 0.5(0.1)(4) + 0.098 = 0.05v_2^2 + 0.049 0.2 + 0.098 = 0.05v_2^2 + 0.049 0.298 = 0.05v_2^2 + 0.049 0.05v_2^2 = 0.298 - 0.049 0.05v_2^2 = 0.249 v_2^2 = (0.249)/(0.05) v_2^2 = 4.98 m^2/s^2 Step 5: Calculate v_2. v_2 = sqrt(4.98 m)^2/s^2 v_2 ≈ 2.23 m/s The speed of the pendulum bob when it is 50 mm above its lowest point is 2.23 m/s. Question 6: Figure 9 shows a roller coaster of mass 500 kg rolling down a section of track. At point A its speed is 10 m/s and its height above the lowest point on the track is 30 m. Point B is 15 m above the lowest point. Show by calculation whether the roller coaster has sufficient mechanical energy to reach point B. (Assume that no energy is lost in overcoming friction.) Step 1: Calculate the total mechanical energy (ME_A) of the roller coaster at point A. ME_A = KE_A + GPE_A ME_A = (1)/(2)mv_A^2 + mgh_A ME_A = (1)/(2)(500 kg)(10 m/s)^2 + (500 kg)(9.8 m/s^2)(30 m) ME_A = (1)/(2)(500)(100) + 147000 ME_A = 25000 J + 147000 J ME_A = 172000 J Step 2: Calculate the gravitational potential energy (GPE_B) required for the roller coaster to reach point B. GPE_B = mgh_B GPE_B = (500 kg)(9.8 m/s^2)(15 m) GPE_B = 73500 J Step 3: Compare the mechanical energy at point A with the potential energy required to reach point B. Since ME_A = 172000 J and GPE_B = 73500 J, and ME_A > GPE_B, the roller coaster has sufficient mechanical energy to reach point B. It will have kinetic energy remaining at point B. The roller coaster has sufficient mechanical energy to reach point B. That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-08-10T17:30:51.598Z

Here are the calculations for the questions: Question 4: A trolley of mass 1 kg rolls down an inclined plane. The trolley on the inclined plane has an initial velocity of 0.2 m/s. When it reaches the bottom its velocity is 2.2 m/s. 4.1 Calculate h, the change in its vertical height. Step 1: Apply the principle of conservation of mechanical energy. Let the initial height be h and the final height be 0 m. (1)/(2)mv_i^2 + mgh_i = (1)/(2)mv_f^2 + mgh_f Step 2: Substitute the given values (m = 1 kg, v_i = 0.2 m/s, h_i = h, v_f = 2.2 m/s, h_f = 0 m, g = 9.8 m/s^2). (1)/(2)(1 kg)(0.2 m/s)^2 + (1 kg)(9.8 m/s^2)h = (1)/(2)(1 kg)(2.2 m/s)^2 + (1 kg)(9.8 m/s^2)(0 m) Step 3: Simplify and solve for h. 0.5(0.04) + 9.8h = 0.5(4.84) + 0 0.02 + 9.8h = 2.42 9.8h = 2.42 - 0.02 9.8h = 2.40 h = (2.40)/(9.8) h ≈ 0.24 m The change in vertical height is 0.24 m. 4.2 Analyse the equation and calculation that you used to answer Question 4.1 and explain whether you needed to know the mass of the trolley to do the calculation. The equation used is the conservation of mechanical energy: (1)/(2)mv_i^2 + mgh_i = (1)/(2)mv_f^2 + mgh_f. If we divide every term by the mass m, the equation becomes (1)/(2)v_i^2 + gh_i = (1)/(2)v_f^2 + gh_f. This shows that the mass m cancels out. Therefore, you did not need to know the mass of the trolley to calculate the change in vertical height. 4.3 Given the same situation and the same initial and final velocities, would the change in vertical height be the same for a 2 kg trolley? Yes, the change in vertical height would be the same for a 2 kg trolley. As explained in 4.2, the mass cancels out in the conservation of mechanical energy equation when calculating height change based on velocity changes. Question 5: Figure 8 shows a pendulum bob of mass 100 g that is 100 mm above its lowest point and has a speed of 2 m/s. Calculate its speed when it is 50 mm above its lowest point. Step 1: Convert units to SI units. m = 100 g = 0.1 kg h_1 = 100 mm = 0.1 m h_2 = 50 mm = 0.05 m Step 2: Apply the principle of conservation of mechanical energy. (1)/(2)mv_1^2 + mgh_1 = (1)/(2)mv_2^2 + mgh_2 Step 3: Substitute the known values (m = 0.1 kg, v_1 = 2 m/s, h_1 = 0.1 m, h_2 = 0.05 m, g = 9.8 m/s^2). (1)/(2)(0.1 kg)(2 m/s)^2 + (0.1 kg)(9.8 m/s^2)(0.1 m) = (1)/(2)(0.1 kg)v_2^2 + (0.1 kg)(9.8 m/s^2)(0.05 m) Step 4: Simplify and solve for v_2^2. 0.5(0.1)(4) + 0.098 = 0.05v_2^2 + 0.049 0.2 + 0.098 = 0.05v_2^2 + 0.049 0.298 = 0.05v_2^2 + 0.049 0.05v_2^2 = 0.298 - 0.049 0.05v_2^2 = 0.249 v_2^2 = (0.249)/(0.05) v_2^2 = 4.98 m^2/s^2 Step 5: Calculate v_2. v_2 = sqrt(4.98 m)^2/s^2 v_2 ≈ 2.23 m/s The speed of the pendulum bob when it is 50 mm above its lowest point is 2.23 m/s. Question 6: Figure 9 shows a roller coaster of mass 500 kg rolling down a section of track. At point A its speed is 10 m/s and its height above the lowest point on the track is 30 m. Point B is 15 m above the lowest point. Show by calculation whether the roller coaster has sufficient mechanical energy to reach point B. (Assume that no energy is lost in overcoming friction.) Step 1: Calculate the total mechanical energy (ME_A) of the roller coaster at point A. ME_A = KE_A + GPE_A ME_A = (1)/(2)mv_A^2 + mgh_A ME_A = (1)/(2)(500 kg)(10 m/s)^2 + (500 kg)(9.8 m/s^2)(30 m) ME_A = (1)/(2)(500)(100) + 147000 ME_A = 25000 J + 147000 J ME_A = 172000 J Step 2: Calculate the gravitational potential energy (GPE_B) required for the roller coaster to reach point B. GPE_B = mgh_B GPE_B = (500 kg)(9.8 m/s^2)(15 m) GPE_B = 73500 J Step 3: Compare the mechanical energy at point A with the potential energy required to reach point B. Since ME_A = 172000 J and GPE_B = 73500 J, and ME_A > GPE_B, the roller coaster has sufficient mechanical energy to reach point B. It will have kinetic energy remaining at point B. The roller coaster has sufficient mechanical energy to reach point B. That's 2 down. 3 left today — send the next one.

1 Calculate h, the change in its vertical height. 4.2 Analyse the equation and calculation that you used to answer Question 4.1 and explain whether you needed to know the mass of the trolley to do the calculation.

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1 Calculate h, the change in its vertical height. 4.2 Analyse the equation and calculation that you used to answer Question 4.1 and explain whether you needed to know the mass of the trolley to do the calculation.

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