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AP Physics 1: Algebra-Based
Unit 3: Work, Energy, and Power
3.2 Kinetic Energy and the Work-Energy Theorem
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What is kinetic energy defined as?
Energy due to motion
In the kinetic energy formula, 'm' represents mass and 'v' represents
velocity
.
True
What does the work-energy theorem state?
Net work equals change in kinetic energy
Kinetic energy is the energy of
motion
What is the starting point for deriving the kinetic energy formula?
Definition of work
Steps to derive the kinetic energy formula
1️⃣ Start with work definition:
W
=
W =
W
=
∫
F
d
x
\int F dx
∫
F
d
x
2️⃣ Use Newton's second law:
F
=
F =
F
=
m
a
ma
ma
3️⃣ Substitute and express acceleration in terms of velocity:
a
=
a =
a
=
d
v
d
t
\frac{dv}{dt}
d
t
d
v
4️⃣ Change variables and integrate:
W
=
W =
W
=
∫
m
v
d
v
\int mv \, dv
∫
m
v
d
v
5️⃣ Integrate from
v
1
v_{1}
v
1
to
v
2
v_{2}
v
2
:
W
=
W =
W
=
1
2
m
(
v
2
2
−
v
1
2
)
\frac{1}{2}m(v_{2}^{2} - v_{1}^{2})
2
1
m
(
v
2
2
−
v
1
2
)
How is work defined in physics?
Energy transferred by a force
Work is only done when the force and displacement are in the same direction.
True
What factors does the kinetic energy of an object depend on?
Mass and velocity
The kinetic energy of an object increases with the square of its
velocity
The work-energy theorem relates net work to the change in
kinetic energy
.
True
Kinetic energy is the energy of
motion
The formula for kinetic energy is K = \frac{1}{2}
mv^{2}
</latex>
True
What does the Work-Energy Theorem relate?
Net work and kinetic energy
Steps in deriving the kinetic energy formula
1️⃣ Start with the definition of work:
W
=
W =
W
=
∫
F
d
x
\int F dx
∫
F
d
x
2️⃣ Use Newton's second law:
F
=
F =
F
=
m
a
ma
ma
3️⃣ Substitute:
W
=
W =
W
=
∫
m
a
d
x
\int ma dx
∫
ma
d
x
4️⃣ Express acceleration in terms of velocity:
a
=
a =
a
=
d
v
d
t
\frac{dv}{dt}
d
t
d
v
5️⃣ Substitute and change variables:
W
=
W =
W
=
∫
m
v
d
v
\int mv \, dv
∫
m
v
d
v
6️⃣ Integrate from
v
1
v_{1}
v
1
to
v
2
v_{2}
v
2
:
W
=
W =
W
=
1
2
m
(
v
2
2
−
v
1
2
)
\frac{1}{2} m(v_{2}^{2} - v_{1}^{2})
2
1
m
(
v
2
2
−
v
1
2
)
7️⃣ Define kinetic energy:
K
=
K =
K
=
1
2
m
v
2
\frac{1}{2}mv^{2}
2
1
m
v
2
Newton's second law states that
F
=
F =
F
=
m
a
ma
ma
True
The integral of
∫
m
v
d
v
\int mv \, dv
∫
m
v
d
v
from v_{1}</latex> to
v
2
v_{2}
v
2
equals \frac{1}{2}m(v_{2}^{2} - v_{1}^{2})
What is the change in kinetic energy of a 5 kg object accelerating from 0 m/s to 10 m/s?
250 J
The expression for acceleration in terms of velocity is
a
=
a =
a
=
d
v
d
t
\frac{dv}{dt}
d
t
d
v
True
Match the quantity with its formula:
Work ↔️
W
=
W =
W
=
F
d
cos
θ
Fd\cos\theta
F
d
cos
θ
Net Work ↔️
W
n
e
t
=
W_{net} =
W
n
e
t
=
Δ
K
\Delta K
Δ
K
Kinetic Energy ↔️
K
=
K =
K
=
1
2
m
v
2
\frac{1}{2}mv^{2}
2
1
m
v
2
What is work defined as?
Energy transferred by force
The formula for work is
W
=
W =
W
=
F
d
cos
θ
Fd\cos\theta
F
d
cos
θ
, where θ is the angle between the force and the displacement
Work is only done when the force and
displacement
are in the same direction.
True
What are the units of work?
Joules
The Work-Energy Theorem states that
W
n
e
t
=
W_{net} =
W
n
e
t
=
Δ
K
\Delta K
Δ
K
True
The Work-Energy Theorem mathematically relates the net work to the change in
kinetic
energy.
If a 2 kg object has an initial kinetic energy of 100 J and a net work of 50 J is done on it, what is its final kinetic energy?
150 J
The Work-Energy Theorem can be derived using Newton's second law and the
definition of work
.
True
Substituting
F
=
F =
F
=
m
a
ma
ma
into the work integral gives
W
=
W =
W
=
∫
m
a
d
x
\int ma \, dx
∫
ma
d
x
, which can be simplified using velocity
What does the final step in the derivation of the Work-Energy Theorem conclude?
W
n
e
t
=
W_{net} =
W
n
e
t
=
Δ
K
\Delta K
Δ
K
The work-energy theorem states that the net work done on an object is equal to the change in its
kinetic
energy.
What is the formula for the work-energy theorem?
W
n
e
t
=
W_{net} =
W
n
e
t
=
Δ
K
\Delta K
Δ
K
The work-energy theorem relates forces to changes in
kinetic energy
.
True
What is the formula for kinetic energy?
K
=
K =
K
=
1
2
m
v
2
\frac{1}{2}mv^{2}
2
1
m
v
2
Match the concept with its definition:
Kinetic Energy ↔️ Energy of motion
Work-Energy Theorem ↔️ Net work equals change in kinetic energy
What are the units of kinetic energy?
Joules
Steps to derive the kinetic energy formula
1️⃣ Start with the definition of work
2️⃣ Use Newton's second law
3️⃣ Substitute to express work
4️⃣ Express acceleration in terms of velocity
5️⃣ Substitute and change variables
6️⃣ Integrate to find work
Work is the energy transferred when a force causes an object to move a certain
distance
.
What is the formula for work done when a force acts at an angle to displacement?
W = Fd\cos\theta</latex>
Match the variable in the work formula with its unit:
W ↔️ Joules (J)
F ↔️ Newtons (N)
d ↔️ Meters (m)
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