Work Energy and Power

Step 1: Write the equation for power
Power (P) is force (F) multiplied by velocity (v) :
P=F⋅v
Step 2: Replace force with mass and acceleration
Force is mass (m) times acceleration (‌

dv

dt

) :
P=m‌

dv

dt

⋅v
Step 3: Rearrange to separate variables
Move terms so you have all v terms on one side and t terms on the other:
v‌

dv

dt

=‌

P

m

Step 4: Integrate both sides
Integrate with respect to v and t :
∫vdv=‌

P

m

‌∫dt
After integrating:
‌

v2

2

=‌

P

m

t
Step 5: Solve for velocity (v)
v=√‌

2Pt

m

Step 6: Find the relationship between displacement and time
Velocity is the rate of change of displacement, so:
‌

ds

dt

=√‌

2Pt

m

Step 7: Integrate to get displacement (s)
Integrate both sides with respect to t :
∫ds=√‌

2P

m

‌∫t1∕2‌dt
The right side integrates to t3∕2 :
s∝t3∕2
Step 8: Answer
The value of x is ‌

3

2

.