What minimum speed does a 95.9 g puck need to make it to the top of a 2.7-m-long, 15° frictionless ramp?
min KE=PE at top
1/2 m v^2=mgh
v^2=2g*2.7sin15
To find the minimum speed needed for the puck to make it to the top of the ramp, we can use the principles of energy conservation. The minimum speed required will be when all of its initial kinetic energy is converted to potential energy at the top of the ramp.
Let's break down the problem step by step:
Step 1: Calculate the potential energy at the top of the ramp.
The potential energy (PE) is given by the equation: PE = m * g * h, where m is the mass of the puck, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height the puck reaches at the top of the ramp.
Since the puck is on a ramp inclined at 15°, the vertical height can be calculated as follows:
h = length of the ramp * sin(θ), where θ is the angle of the ramp in radians.
First, convert the angle from degrees to radians:
θ (in radians) = 15° * (π/180)
Now we can calculate the height:
h = 2.7 m * sin(15° * (π/180))
Step 2: Calculate the initial kinetic energy.
The initial kinetic energy (KE) is given by the equation: KE = 1/2 * m * v^2, where v is the initial velocity of the puck.
Since we are looking for the minimum speed, we can assume the puck starts from rest and has an initial velocity of 0 m/s.
Therefore, the initial kinetic energy is: KE = 0
Step 3: Equate the initial kinetic energy to the potential energy.
Since energy is conserved, the initial kinetic energy is equal to the potential energy at the top of the ramp:
KE = PE
0 = m * g * h
Step 4: Solve for the minimum speed.
Rearrange the equation to solve for the minimum speed (v):
v = √(2 * g * h)
Substitute the values into the equation:
v = √(2 * 9.8 m/s^2 * (2.7 m * sin(15° * (π/180))))
Now, you can solve for v using a scientific calculator or using software like Python, MATLAB, or Microsoft Excel.
Remember to include the appropriate units in your answer.
Please note that this calculation assumes a frictionless ramp and neglects any other forces acting on the puck.