Questions: As the block slides down the ramp, what happens to its kinetic energy K, potential energy U, and total mechanic K decreases; U increases; E stays the same K decreases; U increases; E increases K increases; U increases; E increases K increases; U decreases; E stays the same Using conservation of energy, find the speed vb of the block at the bottom of the ramp. Express your answer in terms of some or all the variables m, v, and h and any appropriate constants.

As the block slides down the ramp, what happens to its kinetic energy K, potential energy U, and total mechanic

K decreases; U increases; E stays the same
K decreases; U increases; E increases
K increases; U increases; E increases
K increases; U decreases; E stays the same

Using conservation of energy, find the speed vb of the block at the bottom of the ramp. Express your answer in terms of some or all the variables m, v, and h and any appropriate constants.

Transcript text: As the block slides down the ramp, what happens to its kinetic energy $K$, potential energy $U$, and total mechanic $K$ decreases; $U$ increases; $E$ stays the same $K$ decreases; $U$ increases; $E$ increases $K$ increases; $U$ increases; $E$ increases $K$ increases; $U$ decreases; $E$ stays the same Submit Previous Answers Correct Part G Using conservation of energy, find the speed $v_{\mathrm{b}}$ of the block at the bottom of the ramp. Express your answer in terms of some or all the variables $m, v$, and $h$ and any appropriate constants. View Available Hint(s) Hint 1. How to approach the problem Submit Previous Answers Request Answer Incorrect; Try Again; 4 attempts remaining Part H Complete previous part(s) Part I Complete previous part(s) Part J Complete previous part(s)

Solution

Solution Steps

Step 1: Identify the Given Information

We need to find the speed $ v_b $ of the block at the bottom of the ramp using conservation of energy. The variables involved are mass $ m $, gravitational acceleration $ g $, height $ h $, and initial speed $ v_i $.

Step 2: Apply Conservation of Energy Principle

The total mechanical energy at the top of the ramp (potential energy + kinetic energy) should equal the total mechanical energy at the bottom of the ramp (kinetic energy).

Step 3: Write the Energy Equations

At the top of the ramp: \[ E_{\text{top}} = U_{\text{top}} + K_{\text{top}} = mgh + \frac{1}{2}mv_i^2 \]

At the bottom of the ramp: \[ E_{\text{bottom}} = K_{\text{bottom}} = \frac{1}{2}mv_b^2 \]

Step 4: Set Up the Conservation of Energy Equation

Since energy is conserved: \[ mgh + \frac{1}{2}mv_i^2 = \frac{1}{2}mv_b^2 \]

Step 5: Solve for $ v_b $

\[ mgh + \frac{1}{2}mv_i^2 = \frac{1}{2}mv_b^2 \] \[ 2mgh + mv_i^2 = mv_b^2 \] \[ 2gh + v_i^2 = v_b^2 \] \[ v_b = \sqrt{2gh + v_i^2} \]