Kinematics

Question 1

How to find instantaneous velocity

Answer 1

v=ds/dt
Change in displacement with respect to time

Question 2

Instantaneous acceleration

Answer 2

a=dv/dt
Change in velocity with respect to time

Question 3

How to find instantaneous acceleration when displacement is given

Answer 3

a = dv/dt
v = ds/dt
sub in for v
a = d/dv* ds/dt
a = d^2s/dt^2
Second order differentiation of s with respect to t

Question 4

How to derive v= u +at

Answer 4

a = dv/dt
Rearrange so
dv =adt
Add integrals and limits
v/u§dv = t/t=0§adt
Bring out a from integral
v/u§1dv = a t/t=0§1dt
Integrate both sides
[v]v/u = a [t] t/t=0
(v)-(u) = a(t) - a(0)
v-u= at
v= u + at

Question 5

How to derive s=ut+1/2at^2

Answer 5

v=ds/dt v= u +at
Sub in
u + at = ds/dt
(u+at)dt=1ds
Add Integrate both sides with limits
t/t=0§(u+at)dt=s/s=0§1ds
Integrate
[ut+1/2at^2]t/t=0= [s] s/s=0
(ut+1/2at^2) - (u(0)+ 1/2a(0)^2)= (s) - (0)
s = ut +1/2at^2

Question 6

How to derive v^2 = u^2 + 2as

Answer 6

v=u + at s = ut +1/2at^2
t = (v-u)/a
Sub in for t
Multiply both sides by 2a and remember how the fraction will cancel out.
Solve for v^2= u^2 + 2as

Question 7

When solving equations with calculus methods:

Answer 7

Always include limits where necessary.
When solving use equations given and rearrange substitue in and solve.
First line should be basic relationship found in data booklet. eg, a=dv/dt