for full credit show all work, use figs, and box final answers.
multiple choice question are 2 points each, total 10 points
1. in which one of the following systems is there a decrease in gravitational potential energy?
a. a boy stretches a horizontal spring
b. a car ascends a steep hill
c. a girl jumps down from a bed
d. water is forced upward through a pipe
e. a crate rests at the bottom of an inclined plane
2. a bicyclist is traveling at a speed of 20.0 m/s as he approaches the bottom of a hill. he decides
to coast up the hill and stop upon reaching the top. neglecting friction, determine the vertical
height of the hill.
a. 28.5 m
b. 3.7 m
c. 11.2 m
d. 40.8 m
e. 20.4 m
3. a care package is dropped from rest from a helicopter hovering 25 m above the group. what is
the speed of the package just before it reaches the ground? neglect air resistance.
a. 22 m/s
b. 16 m/s
c. 12 m/s
d. 8.0 m/s
e. 5.0 m/s
4. you throw a baseball straight up. compare the sign of the work done by gravity while
the ball goes up with the sign of the work done by gravity while it goes down.
a. the work is positive on the way up and positive on the way down.
b. the work is positive on the way up and negative on the way down.
c. the work is negative on the way up and positive on the way down.
d. the work is negative on the way up and on the way down because gravity is always
downward.
5. a stone is held at a height ( h ) above the ground. a second stone with four times the mass
of the first one is held at the same height. the gravitational potential energy of the
second stone compared to that of the first stone is
a. one - fourth as much.
b. one - half as much.
c. twice as much.
d. four times as much.
e. the same.

Question

for full credit show all work, use figs, and box final answers.
multiple choice question are 2 points each, total 10 points
1. in which one of the following systems is there a decrease in gravitational potential energy?
   a. a boy stretches a horizontal spring
   b. a car ascends a steep hill
   c. a girl jumps down from a bed
   d. water is forced upward through a pipe
   e. a crate rests at the bottom of an inclined plane
2. a bicyclist is traveling at a speed of 20.0 m/s as he approaches the bottom of a hill. he decides
to coast up the hill and stop upon reaching the top. neglecting friction, determine the vertical
height of the hill.
   a. 28.5 m
   b. 3.7 m
   c. 11.2 m
   d. 40.8 m
   e. 20.4 m
3. a care package is dropped from rest from a helicopter hovering 25 m above the group. what is
the speed of the package just before it reaches the ground? neglect air resistance.
   a. 22 m/s
   b. 16 m/s
   c. 12 m/s
   d. 8.0 m/s
   e. 5.0 m/s
4. you throw a baseball straight up. compare the sign of the work done by gravity while
the ball goes up with the sign of the work done by gravity while it goes down.
   a. the work is positive on the way up and positive on the way down.
   b. the work is positive on the way up and negative on the way down.
   c. the work is negative on the way up and positive on the way down.
   d. the work is negative on the way up and on the way down because gravity is always
downward.
5. a stone is held at a height ( h ) above the ground. a second stone with four times the mass
of the first one is held at the same height. the gravitational potential energy of the
second stone compared to that of the first stone is
   a. one - fourth as much.
   b. one - half as much.
   c. twice as much.
   d. four times as much.
   e. the same.

Accepted Answer

### Question 1
# Explanation:
## Step1: Recall Gravitational PE Formula
Gravitational potential energy (PE) is $ PE = mgh $, where $ m $ is mass, $ g $ is gravity, and $ h $ is height. A decrease in $ PE $ occurs when height $ h $ decreases (mass and $ g $ constant).

## Step2: Analyze Each Option
- **Option a**: Stretching a spring involves elastic PE, not gravitational PE. No change in height. Eliminate.
- **Option b**: A car ascending a hill increases height ($ h $ increases), so $ PE $ increases. Eliminate.
- **Option c**: A girl jumping down from a bed decreases her height ($ h $ decreases), so $ PE = mgh $ decreases (since $ h $ is smaller). This fits.
- **Option d**: Water forced upward increases height ($ h $ increases), so $ PE $ increases. Eliminate.
- **Option e**: A crate at the bottom of an incline has low height; resting there means no change in height. $ PE $ is constant. Eliminate.

# Answer: c. A girl jumps down from a bed

### Question 2
# Explanation:
## Step1: Apply Conservation of Energy
Neglecting friction, kinetic energy (KE) at the bottom converts to gravitational potential energy (PE) at the top.
$ KE_{\text{initial}} = PE_{\text{final}} $
$ \frac{1}{2}mv^2 = mgh $

## Step2: Solve for Height $ h $
Mass $ m $ cancels:
$ \frac{1}{2}v^2 = gh $
$ h = \frac{v^2}{2g} $

Substitute $ v = 20.0 \, \text{m/s} $, $ g = 9.8 \, \text{m/s}^2 $:
$ h = \frac{(20.0)^2}{2(9.8)} = \frac{400}{19.6} \approx 20.4 \, \text{m} $? Wait, no—wait, let’s recalculate:
$ 20^2 = 400 $; $ 2 \times 9.8 = 19.6 $; $ 400 \div 19.6 \approx 20.4 $? Wait, no, the options include 20.4? Wait, the options are a.28.5, b.3.7, c.11.2, d.40.8, e.20.4. Wait, let’s check again:

Wait, $ v = 20 \, \text{m/s} $, so $ KE = 0.5 \times m \times 20^2 = 200m $.
$ PE = mgh $, so $ 200m = mgh $ → $ h = 200 / 9.8 ≈ 20.4 \, \text{m} $. So option e.

# Answer: e. 20.4 m

### Question 3
# Explanation:
## Step1: Apply Kinematic Equations or Energy Conservation
Using conservation of energy: Potential energy (PE) at height $ h $ converts to kinetic energy (KE) at the ground.
$ PE_{\text{initial}} = KE_{\text{final}} $
$ mgh = \frac{1}{2}mv^2 $

## Step2: Solve for Speed $ v $
Mass $ m $ cancels:
$ gh = \frac{1}{2}v^2 $
$ v = \sqrt{2gh} $

Substitute $ g = 9.8 \, \text{m/s}^2 $, $ h = 25 \, \text{m} $:
$ v = \sqrt{2(9.8)(25)} = \sqrt{490} \approx 22 \, \text{m/s} $ (since $ \sqrt{490} \approx 22.14 $, closest to 22 m/s).

# Answer: a. 22 m/s

### Question 4
# Explanation:
## Step1: Recall Work Done by Gravity
Work $ W = F \cdot d \cdot \cos\theta $, where $ F $ is force (gravity, $ F = mg $ downward), $ d $ is displacement, and $ \theta $ is the angle between $ F $ and $ d $.

## Step2: Analyze Upward and Downward Motion
- **Upward motion**: Displacement $ d $ is upward, gravity $ F $ is downward. So $ \theta = 180^\circ $, $ \cos\theta = -1 $.
  $ W = mg \cdot d \cdot (-1) = -mgd $ (negative work, since force and displacement oppose).

- **Downward motion**: Displacement $ d $ is downward, gravity $ F $ is downward. So $ \theta = 0^\circ $, $ \cos\theta = 1 $.
  $ W = mg \cdot d \cdot (1) = +mgd $ (positive work, since force and displacement are in the same direction).

# Answer: c. The work is negative on the way up and positive on the way down.

### Question 5
# Explanation:
## Step1: Recall Gravitational PE Formula
Gravitational potential energy is $ PE = mgh $, where $ m $ is mass, $ g $ is gravity, and $ h $ is height.

## Step2: Compare the Two Stones
- First stone: $ PE_1 = m_1gh $
- Second stone: $ m_2 = 4m_1 $, same height $ h $, so $ PE_2 = m_2gh = 4m_1gh = 4PE_1 $.

# Answer: d. four times as much.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Recall Gravitational PE Formula

Step2: Analyze Each Option

Step1: Apply Conservation of Energy

Step2: Solve for Height \( h \)

Step1: Apply Kinematic Equations or Energy Conservation

Step2: Solve for Speed \( v \)

Answer:

Question 2