Final answer:
The minimum speed required for a puck to reach the top of a frictionless ramp is found by using energy conservation, converting kinetic energy at the base into potential energy at the top. The steps include calculating the height of the ramp via trigonometry, computing potential energy with the puck's mass and gravity, and then equating it to the initial kinetic energy to solve for the initial velocity.
Step-by-step explanation:
The student's question asks about the minimum speed required for a puck to reach the top of a frictionless ramp.
To find this, we use the principles of energy conservation, specifically the conversion of kinetic energy to gravitational potential energy.
The energy conservation equation is given by KE_initial + PE_initial = KE_final + PE_final where KE is kinetic energy, and PE is potential energy.
For this scenario, the final kinetic energy at the top of the ramp is zero since the puck is momentarily at rest at the highest point.
Therefore, the initial kinetic energy must equal the potential energy at the top.
Let's break it down step by step:
First, find the height of the ramp by using trigonometry: height = ramp_length * sin(incline_angle).
Calculate the potential energy at the top using the height, mass of the puck, and gravitational acceleration (PE = m * g * h).
Determine the initial kinetic energy required by using the potential energy calculated (KE_initial = PE_final).
Finally, solve for the initial velocity using the kinetic energy formula (KE = 0.5 * m * v^2).
Going through the math:
Given the ramp length (L) is 3.4 m and the incline angle (theta) is 29 degrees, we find the height (h = L * sin(theta)).
With the height known, use PE = m * g * h to calculate potential energy.
Set KE_initial = PE, since at the top KE_final is zero (no kinetic energy).
From KE_initial, rearrange 0.5 * m * v^2 = PE to solve for the velocity (v).
The numbers will provide the minimum speed needed for the puck to reach the top of the ramp.