Dynamics & Relativity 1 Flashcards
for uniform motion r(t)=
r(t0)+v(to)(t-t0)+1/2A(t-t0)^2
velocity
defined as rate of change of positions
v(t) = dr/dt = dxp/dt i + dyp/dt j + dzp/dt k
acceleration
defined as the rate of change of velocity
a(t)=dv/dt=d^2r/dt^2 (same idea in vector components as velocity but with second derivative)
simpler notation for time dependence
r0 = r(t0)
r = r(t)
hence r=ro+vo(t-t0)+1/2a(t-t0)^2
motion in 1 dimension
pick just a single direction eg x
r=xi, v=vxi, a=axi
vx=dx/dt, ax=d&2x/dt^2 = constant
derivation of vx=vx0+ax(t-t0) using integration
start with ax=dvx/dt and integration both sides wrt time
on lhs, a constant so get ax(t-t0)
rhs gives vx-vx0
put together a rearrange
derivation of x-x0=vx0(t-t0)+ax/2(t-t0)^2
use vx=dx/dt
integrate both sides wrt time
insert previous equation for velocity and rearrange
how to make x-x0=vx0(t-t0)+ax/2(t-t0)^2 more familiar?
setting initial position and time to zero gives
x=vx0t+1/2axt^2
derivation of vx^2=vx0^2+2ax(x-x0)
combining previous two equations and eliminating (t-t0)
derivation of x-x0=1/2(vx+vx0)(t-t0)
combining previous two equations and eliminating acceleration
assumptions for free falling bodies
- gravitational acceleration due to Earth’s gravity is constant
- ignore gravity from everything but Earth
- ignore rotation of Earth
- Pretend Earth is flat
- Ignore air resistance
free falling body setup
ay=-g
vy=vy0-g(t-t0)
y-y0=vy0(t-t0)-1/2g(t-t0)^2
free falling bodies - things to check
- units
- signs (heights +Ve or 0, object moving down so y-component of velocity is -ve
3.magnitudes (timescale a few secodns, distance few tens of metres etc)
motion in 2 dimensions
study the motion in each dimension seperately
acceleration in x direction
0
hence vx=vx0
separating velocity into components
vx0=v0costheta
vy0=v0sintheta
how to find y as a function of x
take equation for y and use equation for x to eliminate t
how far does an object travel (x direction)
set vertical position to zero and rearrange for Xr
how high does object travel?
equations of motion for vy to find time when vy=0
or find dy/dx and set to 0 to find x at highest point. Use this to find y at highest point
or use symmetry if highest point is half of journey.
maximum height only depends on
g and vy0
relative motion
r p/a = r b/a + r p/b (similar for relative velocity if find derivative)
“the velocity of the point 𝑃
in coordinate system 𝐴 is equal to the
velocity of 𝑃 in coordinate system 𝐵 plus
the velocity of the origin of system 𝐵 as
measured in 𝐴”
gradient of x-t plot
vx
polar coordinates
r, theta
r=(x^2+y^2)^1/2
theta=tan^-1(y/x)
w, angular spped
rate of change of theta
w=d theta/dt
(rad per second)
