Question

On August \(2,1971,\) Astronaut David Scott, while standing on the surface of the Moon, dropped a 1.3 -kg hammer and a 0.030 -kg falcon feather from a height of \(1.6 \mathrm{~m} .\) Both objects hit the Moon's surface \(1.4 \mathrm{~s}\) after being released. What is the acceleration due to gravity on the surface of the Moon?

Short answer

Answer: The acceleration due to gravity on the Moon's surface is approximately 1.63 m/s².

Step-by-step solution

Identify the relevant kinematic equation

We can use the following kinematic equation for uniformly accelerated motion: d = vi*t + 0.5*a*t^2 where d is the distance traveled, vi is the initial velocity (which is 0 in this case since the objects are dropped), a is the acceleration due to gravity on the Moon, and t is the time taken for the objects to fall.

Set up the equation with given values

Because the initial velocity (vi) is 0, the equation simplifies to: d = 0.5*a*t^2 Now, we can plug in the values given in the problem: 1.6 = 0.5*a*(1.4)^2

Solve for acceleration due to gravity (a)

We will now solve the equation for a, the acceleration due to gravity on the Moon: 1.6 = 0.5*a*(1.96) 1.6 = 0.98*a a = 1.6 / 0.98 a ≈ 1.63 m/s^2

Report the answer

The acceleration due to gravity on the surface of the Moon is approximately 1.63 m/s².

Key concepts

Kinematic Equations

Kinematic equations are the cornerstone of classical mechanics and essential for understanding motion. They allow us to predict the future position, velocity, and acceleration of an object when its motion is undergoing a constant acceleration. In our textbook exercise, the motion is vertical and the acceleration is due to gravity on the Moon. One of the fundamental kinematic equations used is \( d = v_i t + \frac{1}{2}a t^2 \), where \( d \) is the displacement, \( v_i \) the initial velocity, \(a\) the acceleration, and \(t\) the time. For objects dropped from rest, \( v_i = 0 \), simplifying our equation.

Understanding this equation is critical for solving problems involving uniformly accelerated motion and free fall, as it encapsulates the relationship between displacement, initial velocity, time, and acceleration. In our moon exercise, we rearranged this equation to solve for the acceleration \(a\), showing students the practical application of kinematic equations in real-world scenarios.

Uniformly Accelerated Motion

Uniformly accelerated motion describes any scenario where an object's velocity changes at a constant rate. This is the kind of motion that occurs in free fall, where gravity is the sole force acting on the object, causing it to accelerate uniformly. On Earth, this acceleration is approximately \(9.81 m/s^2\), but on the Moon, it's significantly less due to the Moon's smaller mass and gravitational force.

In the example from the textbook, the hammer and feather demonstrate uniformly accelerated motion under the Moon's gravity. By measuring the time it takes for them to fall a known distance, we can determine the acceleration due to Moon's gravity. This concept allows us to predict how any object will move under similar conditions, be it on the Moon, Earth, or any other celestial body with a gravitational field.

Free Fall

Free fall occurs when the only force acting on an object is gravity; this makes it a fascinating case study of uniformly accelerated motion. Objects in free fall, regardless of their mass, will have the same acceleration given that no other forces are acting upon them. This was famously demonstrated by Galileo and was again shown to be true during the Apollo 15 mission when an astronaut on the Moon dropped a hammer and a feather simultaneously, and they hit the ground together, just like in our textbook exercise.

In the absence of air resistance—such as on the Moon, where there's no atmosphere—objects will fall at the same rate. This notion is counter-intuitive when we observe objects falling on Earth since air resistance plays a significant role. The concept of free fall not only helps students to grasp the basics of gravitational acceleration but also underscores the universality of physical laws in different environments.