Find an expression for v , the magnitude of the velocity of the top of the object's shadow, at time t .?

Question

A small source of light S is located at a distance L from a vertical wall. An opaque object with a height of h moves toward the wall with constant velocity of magnitude v. At time t=0, the object is located at the source S.

Answer 1

I assume that the light source is at ground level.
At time t the object is d1 from the light source, the top of the shadow is at position p1 on the wall. At time t+dt, the object is d2 from the light source and the top of the shadow is at position p2 on the wall.
d1=v*t
and
d2=v*(t+dt)

vp(t)=(p2-p1)/dt

using a bit of trig
at time t, the angle from the top of the object, h and the source is tan(th1)=h/d1 which is also p1/L
therefore
p1=h*L/d1
similarly
tan(th2)=h/d2=p2/L
so
p2=h*L/d2
so (p2-p1)/dt=
h*L*(1/d2-1/d1)/dt
plugging in d1 and d2
h*L*(1/(t+dt)-1/t)/(v*dt)
simplify
h*L*(t-(t+dt))/(v*dt*t*(t+dt))
keep going
h*L/(v*t*(t+dt))

now integrate from 0 to t

j