Using a spring gun, a 4.0-kg steel block is launched up a lubricated ramp made out of steel. The angle of the ramp was measured to be 38.8º. The inital speed of the block was 12.0 m/s.
(a) Determine the vertical height that the block reaches above its launching point.
(b) What speed does the block have when it sldes back to its startin point?
Vo = 12m/s @ 38.8 Deg.
Xo = 12cos38.8 = 9.35m/s.
Yo = 12sin38.8 = 7.5m/s.
a. h = (Yf^2 - Yo^2) / 2g,
h = (0 - (7.5)^2) / -19.6 = 2.88m.
b. Yf^2 = Yo^2 + 2g*d,
Yf^2 = 0 + 19.6*2.88 = 56.4,
Yf = 7.5m/s.
To solve this problem, we need to consider the conservation of mechanical energy. We know that the initial energy of the block is equal to the final energy of the block. The initial energy consists of the potential energy and the kinetic energy, while the final energy consists of only potential energy.
Let's break it down step by step:
(a) Determine the vertical height that the block reaches above its launching point:
1. Calculate the initial kinetic energy:
Kinetic energy (KE) = (1/2) * mass * velocity²
Substitute the values: KE = (1/2) * 4.0 kg * (12.0 m/s)²
2. Calculate the initial potential energy:
Potential energy (PE) = mass * acceleration due to gravity * height
Since the block is launched vertically, the angle of the ramp is not involved.
PE = 4.0 kg * 9.8 m/s² * height
3. Since the block reaches its highest point, its final kinetic energy is zero. We can set the initial energy equal to the final energy:
Initial kinetic energy + Initial potential energy = Final potential energy
(1/2) * 4.0 kg * (12.0 m/s)² + 4.0 kg * 9.8 m/s² * height = 0
Solve the equation for height:
(1/2) * 4.0 kg * (12.0 m/s)² = -4.0 kg * 9.8 m/s² * height
height = [(1/2) * 4.0 kg * (12.0 m/s)²] / [-4.0 kg * 9.8 m/s²]
Calculate the value to find the vertical height.
(b) Determine the speed of the block when it slides back to its starting point:
To find the speed of the block when it slides back to its starting point, we can use the principle of conservation of mechanical energy. At its lowest point, the potential energy will be zero. The energy at the starting point will be entirely kinetic energy.
1. Calculate the final kinetic energy when the block slides back down the ramp:
Final kinetic energy (KE) = (1/2) * mass * velocity²
KE = (1/2) * 4.0 kg * velocity²
2. Set the final kinetic energy equal to the initial kinetic energy:
Initial kinetic energy = Final kinetic energy
(1/2) * 4.0 kg * (12.0 m/s)² = (1/2) * 4.0 kg * velocity²
Solve the equation for velocity:
(1/2) * 4.0 kg * (12.0 m/s)² = (1/2) * 4.0 kg * velocity²
Calculate the value to find the speed when it slides back to its starting point.
Remember to use the correct units and follow the proper order of operations in your calculations.