Question

The acceleration due to gravity on the surface of Mars at the Equator is \(3.699 \mathrm{~m} / \mathrm{s}^{2}\). A rock released from rest takes \(0.8198 \mathrm{~s}\) to reach the surface. From what height was the rock dropped?

Short answer

Based on the given acceleration due to gravity on Mars and the time it takes for a rock to reach the surface when dropped from rest, the rock was dropped from approximately 1.2347 meters above the surface of Mars at the Equator.

Step-by-step solution

Write down the given values

We are given the values of \(a\), the acceleration due to gravity on Mars, and \(t\), the time taken for the rock to fall to the surface. They are as follows: - \(a = 3.699 \mathrm{~m} / \mathrm{s}^{2}\) - \(t = 0.8198 \mathrm{~s}\)

Use the equation of motion

The equation we are using is \(h_0 = -\frac{1}{2}at^2\) since \(v_0 = 0\) and \(h=0\) as the rock is dropped from rest and falls to the surface. Plug the given values of \(a\) and \(t\) into this equation: \(h_0 = -\frac{1}{2}(3.699 \mathrm{~m} / \mathrm{s}^{2})(0.8198 \mathrm{~s})^2\)

Solve for \(h_0\)

To find \(h_0\), perform the necessary calculations: \(h_0 = -\frac{1}{2}(3.699)(0.8198)^2 \approx -1.2347 \mathrm{~m}\) Since the height cannot be a negative value, we take the absolute value of the result: \(h_0 \approx 1.2347 \mathrm{~m}\) The rock was dropped from approximately 1.2347 meters above the surface of Mars at the Equator.

Key concepts

Acceleration Due to Gravity

When learning about the mysterious forces of the universe, one of the key concepts students come across is the 'acceleration due to gravity'. This force is what keeps us anchored to the ground and causes objects to fall towards the center of massive bodies, like the Earth or Mars.

This acceleration is different on Mars compared to Earth due to the difference in mass and size of the two planets. On Mars, the acceleration due to gravity is only about 38% of Earth's gravity, which means objects will fall slower on Mars than they would on Earth. In the exercise, gravity's acceleration on Mars' surface at the Equator is given as \(3.699 \mathrm{~m} / \mathrm{s}^{2}\).

To accommodate the exercise improvement advice, adding an example would enhance understanding. For instance, if you were to jump on Mars, you'd be able to jump much higher and stay in the air longer due to this lower acceleration. Understanding this concept is crucial when calculating the movement of objects in Martian gravity.

Equations of Motion

The second core concept that plays a pivotal role in physics problems is the 'equations of motion'. These are formulas that allow us to predict how an object will move under the influence of forces, such as gravity. When an object is dropped, we often assume it is in free fall, which means the only force acting on it is gravity.

One such equation that is relevant to our textbook problem is \(h_0 = -\frac{1}{2}at^2\), where \(h_0\) is the object's initial height, \(a\) is the acceleration due to gravity, and \(t\) is the time the object takes to fall. Note that the initial velocity \(v_0\) is zero because the rock is dropped from rest. It's important for students to understand that these equations are simplifications that assume no air resistance or other forces except gravity.

Using the equation of motion in the solution helps us find the height from which the rock was dropped on Mars. Keeping the explanation simple, imagine dropping a ball from a certain height and timing how long it takes to hit the ground; this equation would allow you to calculate how high the drop was.

Free Fall

The concept of 'free fall' can seem quite thrilling. It refers to the motion of an object where gravity is the only force acting upon it. Take skydiving, for example; before the parachute opens, a skydiver is essentially in free fall.

On Mars, in our textbook scenario, the rock experiences free fall. Despite the planet's lower gravity, the rock still accelerates towards the surface just as objects do on Earth, albeit less intensely. The duration of the fall, given as \(0.8198 \mathrm{~s}\), allows us to calculate using our motion equations exactly how high the rock was before it began its Martian descent.

Understanding free fall is essential for study in various fields such as physics, engineering, and even space exploration. In applying this concept, remembering it assumes no other forces but gravity is acting is important, which isn't always the case in real-world scenarios where air resistance and other factors come into play.