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.
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Morning ankrahp297 — let's get this done. Here are the solutions for the two problems. Problem 1: A 0.5 kg object is thrown upwards with a velocity of 18 m/s. Determine the following: a) the height at which potential energy equals kinetic energy b) the kinetic energy c) the potential energy d) the mechanical energy Given: Mass m = 0.5 kg Initial velocity v_0 = 18 m/s Acceleration due to gravity g = 9.8 m/s^2 Step 1: Calculate the initial kinetic energy and total mechanical energy. The total mechanical energy is conserved. At the point of projection (height h=0), all energy is kinetic. E_total = KE_0 = (1)/(2)mv_0^2 E_total = (1)/(2)(0.5 kg)(18 m/s)^2 E_total = (1)/(2)(0.5 kg)(324 m^2/s^2) E_total = 81 J d) The mechanical energy is the total energy, which is conserved. Mechanical energy = 81 J Step 2: Determine the height where potential energy equals kinetic energy. At this height h, PE = KE. Since E_total = KE + PE, we have E_total = PE + PE = 2PE or E_total = KE + KE = 2KE. So, PE = (1)/(2)E_total and KE = (1)/(2)E_total. PE = mgh (1)/(2)E_total = mgh (1)/(2)(81 J) = (0.5 kg)(9.8 m/s^2)h 40.5 J = (4.9 N)h h = 40.5 J4.9 N h ≈ 8.27 m a) The height at which potential energy equals kinetic energy is: h = 8.27 m b) At this height, the kinetic energy is half of the total mechanical energy. KE = (1)/(2)E_total = (1)/(2)(81 J) KE = 40.5 J c) At this height, the potential energy is also half of the total mechanical energy. PE = (1)/(2)E_total = (1)/(2)(81 J) PE = 40.5 J --- Problem 2: A ball is projected upward. At a certain height, its speed is 12 m/s and its kinetic energy equals its potential energy. Calculate the height at that point. Given: Speed v = 12 m/s at a certain height h. At this height, KE = PE. Acceleration due to gravity g = 9.8 m/s^2. Step 1: Express kinetic and potential energy in terms of height and speed. KE = (1)/(2)mv^2 PE = mgh Step 2: Use the condition KE = PE to find the height. (1)/(2)mv^2 = mgh Cancel out the mass m from both sides: (1)/(2)v^2 = gh Step 3: Solve for height h. h = (v^2)/(2g) Substitute the given values: h = (12 m/s)^22(9.8 m/s^2) h = 144 m^2/s^219.6 m/s^2 h ≈ 7.35 m The height at that point is: h = 7.35 m Send me the next one 📸