7 - linear motion under variable force

Question 1

acceleration with velocity as a function of displacement

Answer 1

a = dv/dx*dx/dt
= v dv/dx

Question 2

time with velocity as a function of displacement

Answer 2

v = dx/dt
∫ dt = ∫1/v dx
t = ∫1/v

Question 3

velocity when aqcceleration is a function of displacement

Answer 3

a = v dv/dx
1/2 v² = ∫ a

Question 4

time with acceleration as a function of velocity

Answer 4

a = dv/dt
∫ dt = ∫1/a dv
t = ∫1/a

Question 5

displacement when acceleration is a function of velocity

Answer 5

a = v dv/dx
∫ dx = ∫v/a dv
x = ∫ v/a

Question 6

formulae used to solve

Answer 6

v = dx/dt
a = dv/dt = d²x/dt²
v = ∫ a dt
x = ∫ v dt

Question 7

force as a function of time

Answer 7

F = ma = m dv/dt
∫ F dt = ∫ m dv
mv = ∫ F

Question 8

velocity when force is a function of time

Answer 8

v = 1/m ∫ F

Question 9

force as a function of displacement

Answer 9

F = ma
F = mv dv/dx
∫ F dx = ∫ mv dv
1/2mv² = ∫ F

Question 10

force as a function of velocity

Answer 10

F = m dv/dt
∫ dt = ∫ m/F dv
t = ∫ m/F
or F = mv dv/dx
∫ dx = ∫ mv/F dv
x = ∫ mv/F

Question 11

particles moving in a straight vertical line

Answer 11

have to also take into account weight when considering forces and acceleration

Question 12

power under a variable force

Answer 12

P = Fv