Kinematic Relationships

Question 1

what is the displacement, s, of a particle?

Answer 1

the length and direction from the origin to the particle.

Question 2

calculus formulas for velocity

Answer 2

v=ds/dt v=∫adt

Question 3

calculus formulas for acceleration

Answer 3

a=dv/dt=d^2s/dt^2

Question 4

calculus formula for displacement

Answer 4

s=∫vdt

Question 5

what does gradient represent on a velocity-time and displacement-time graph?

Answer 5

instantaneous rate of change (of velocity/acceleration)

Question 6

how to find the area under a graph (with limits)

Answer 6

integration

Question 7

how to derive v=u+at

Answer 7

a=d^2s/dt^2
integrate with respect to time
∫adt=∫d^2s/dt^2 dt
integrate both sides
at+k=ds/dt
when t=0, ds/dt=u so k=u
when t=t ds/dt=v
at+u=v
v=u+at

Question 8

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

Answer 8

v= ds/dt = u+at
integrate with respect to time
∫v dt = ∫(u+at) dt
∫ds/dt dt = ∫u dt + ∫at dt
s=ut+1/2at^2+c
when t=0, s=0 so c=0
s=ut+1/2at^2

Question 9

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

Answer 9

combine EOM 1&2
v=u+at
v^2=u^2+2uat+a^2t^2
v^2=u^2+2a(ut+1/2at^2)
v^2=u^2+2as