Physics 325, Fall 2006
Prof. Susan Lamb
Homework Assignment #3
Solutions
1a) We can find the distance to impact by requiring that the projectile follows its usual
parabolic path, and also lands on the straight, sloped line of the hill. Denoting the
x
axis
to be horizontal and the
y
axis to be vertical, the
x
component equation for the impact is
v
0
cos
αt
=
r
cos
β
v
0
sin
αt

1
2
gt
2
=
r
sin
β
where
r
is the distance up the hill we’re solving for. The projectile problem without the
hill is solved by finding the time of flight first, and since we need to eliminate
t
from our
equations, let’s do that here. It’s a lot easier to solve the
x
equation for
t
.
t
=
r
cos
β
v
0
cos
α
We can insert this
t
into the second equation. After a small amount of simplification, we
get
gr
2
cos
2
β
2
v
2
0
cos
2
α
+
r
sin
β

r
cos
β
sin
α
cos
α
= 0
This is quadratic in
r
, but it’s an easy one. We already know
r
= 0 is a solution since
the projectile is on the ground when it is launched. We’re interested in the other solution
r >
0, so you can divide by
r
and solve the remaining linear equation.
r
=
2
v
2
0
cos
2
α
g
cos
2
β
cos
β
sin
α
cos
α

sin
β
We can also write this as
r
=
2
v
2
0
cos
2
α
g
cos
β
(tan
α

tan
β
)
After checking the units, you can also check that this is right by setting
β
= 0 and
checking that
r
(
β
= 0) =
2
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 Fall '09
 Lamb
 mechanics, Force, Work, Cos, 0 g, 0.31%, fij, ﬂight ﬁrst

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