A
boulder
rolls
off
a
cliff
that
is
20
m
above
the
surface
of
a
lake
with
an
initially
horizontal
speed
v0.
What
minimum
initial
speed
must
it
have
to
land
on
the
plain?
Hint:
it
must
just
avoid
striking
the
dam
which
is
100
m
away
from
the
base
of
the
cliff,
as
shown
in
the
Iigure.
Neglect
the
size
of
the
boulder
and
neglect
air
resistance
Hint:
vo *
(time
of
flight)
must
exceed
100 m.
There
is
no
ligure.
To find the minimum initial speed required for the boulder to land on the plain without striking the dam, we can use the principles of projectile motion.
The boulder is launched horizontally from the cliff, so its initial vertical velocity is zero. The only force acting on the boulder in the vertical direction is gravity, which causes it to accelerate downward.
The boulder will travel a horizontal distance of 100 m (since it must avoid striking the dam) and a vertical distance of 20 m (from the top of the cliff to the surface of the lake).
Let's assume g is the acceleration due to gravity, which is approximately 9.8 m/s².
First, we can find the time it takes for the boulder to reach the lake's surface. We can use the equation:
h = (1/2)gt²
Rearranging the equation to solve for time t:
2h/g = t²
t = √(2h/g)
Plugging in the values for h = 20 m and g = 9.8 m/s²:
t = √(2 * 20 / 9.8) ≈ √4.08 ≈ 2.02 seconds
Next, we can find the minimum initial speed v0 needed for the boulder to travel a horizontal distance of 100 m in 2.02 seconds. We can use the equation:
d = v0 * t
Rearranging the equation to solve for v0:
v0 = d / t
Plugging in the values for d = 100 m and t = 2.02 seconds:
v0 = 100 / 2.02 ≈ 49.5 m/s
Therefore, the boulder must have a minimum initial speed of approximately 49.5 m/s to land on the plain without striking the dam.
Note: This calculation neglects the size of the boulder and air resistance, as mentioned in the problem statement.