A robot probe drops a camera off the rim of
a 291 m high cliff on Mars, where the free-fall
acceleration is 3.7 m/s
2
.
Find the velocity with which it hits the
ground.
Answer in units of m/s
Find the time required for the camera to reach
the ground.
Answer in units of s
To find the velocity with which the camera hits the ground, we can use the equation for free-fall motion:
v² = u² + 2as
where:
v is the final velocity (which is what we want to find)
u is the initial velocity (0 m/s, as the camera is dropped)
a is the acceleration (3.7 m/s²)
s is the distance (291 m)
Plugging in the values into the equation, we have:
v² = 0 + (2 * 3.7 * 291)
Simplifying:
v² = 2146.2
Taking the square root of both sides:
v = √2146.2
Using a calculator, we find:
v ≈ 46.35 m/s
So, the velocity with which the camera hits the ground is approximately 46.35 m/s.
To find the time required for the camera to reach the ground, we can use the equation:
s = ut + (1/2)at²
where:
s is the distance (291 m)
u is the initial velocity (0 m/s, as the camera is dropped)
a is the acceleration (3.7 m/s²)
t is the time (what we want to find)
Rearranging the equation, we get:
t² + (2u/a)t - (2s/a) = 0
Plugging in the values, we have:
t² + (2 * 0/3.7)t - (2 * 291/3.7) = 0
Simplifying:
t² - (2 * 291/3.7) = 0
Using the quadratic formula, we can find the value of t. The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / 2a
In our case, a = 1, b = 0, and c = -2 * 291/3.7. Plugging in these values, we have:
t = (-0 ± √(0² - 4 * 1 * (-2 * 291/3.7))) / (2 * 1)
Simplifying further:
t = (√(4 * 2 * 291/3.7)) / 2
t = (√(8 * 291/3.7)) / 2
Using a calculator, we find:
t ≈ 12.13 s
So, the time required for the camera to reach the ground is approximately 12.13 s.