Problem 56 - College Physics

Question

In 2004 two Martian probes successfully landed on the Red Planet. The final phase of the landing involved bouncing the probes until they came to rest (they were surrounded by protective inflated "balloons"). During one of the bounces, the telemetry (electronic data sent back to Earth) indicated that the probe took off at \(25.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(20^{\circ}\) and landed \(110 \mathrm{~m}\) away \((\) and then bounced again \()\) Assuming the landing region was level, determine the acceleration due to gravity near the Martian surface.

Step-by-Step Solution

Answer

The acceleration due to gravity on Mars is approximately \(3.65 \mathrm{~m/s^2}\).

Step 1: Break Down the Horizontal Component

First, we need to determine the horizontal and vertical components of the initial velocity. The horizontal component can be found using the equation: \( v_{x} = v \cos \theta \), where \( v = 25.0 \mathrm{~m/s} \) and \( \theta = 20^{\circ} \). Plugging in these values gives:\[ v_{x} = 25.0 \cos 20^{\circ} \approx 23.49 \mathrm{~m/s} \]

Step 2: Calculate Horizontal Time of Flight

Next, we calculate the time of flight using the horizontal distance. The formula for horizontal distance is: \( x = v_{x} \cdot t \). We know that the horizontal distance traveled, \( x = 110 \mathrm{~m} \), and \( v_{x} = 23.49 \mathrm{~m/s} \), so:\[110 \mathrm{~m} = 23.49 \mathrm{~m/s} \cdot t\]Solving for \( t \), we find:\[t \approx \frac{110}{23.49} \approx 4.68 \mathrm{~s}\]

Step 3: Break Down the Vertical Component

To find the vertical component, use the formula: \( v_{y} = v \sin \theta \). Using the given initial speed \( v = 25.0 \mathrm{~m/s} \) and \( \theta = 20^{\circ} \), we calculate:\[ v_{y} = 25.0 \sin 20^{\circ} \approx 8.55 \mathrm{~m/s} \]

Step 4: Determine Vertical Acceleration (Martian Gravity)

The vertical motion can be described by the equation: \( 0 = v_{y} - g \cdot t \), where \( 0 \) is the final vertical velocity when it reaches the highest point before descending back down. Rearranging gives:\[ t = \frac{v_{y}}{g}\]We know the total time of flight is \(4.68\) seconds, so the time to the peak is half of this, \(t_{peak} = \frac{4.68}{2} \approx 2.34\) seconds. Setting up the equation gives:\[ 2.34 = \frac{8.55}{g} \]Solving for \( g \), we find:\[\ g \approx \frac{8.55}{2.34} \approx 3.65 \mathrm{~m/s^2} \]

Step 5: Verify the Calculation

Recheck each calculation step to ensure accuracy. Verify that the gravity calculation is consistent with the formula and that the time of flight and other derived values match the known conditions (i.e., distance and velocities). Confirm that everything is calculated correctly for the Martian conditions.

Key Concepts

Martian Gravity

Martian gravity plays a crucial role in understanding the physics of motion on Mars. The gravitational acceleration on Mars is weaker than on Earth due to its smaller size and mass. While Earth's gravity pulls objects at about \(9.8 \, \mathrm{m/s^2}\), Martian gravity is roughly \(3.7 \, \mathrm{m/s^2}\). This reduced force affects how objects move and land on the Martian surface.

When solving physics problems related to Martian gravity, it's important to adjust our expectations. For example, if a probe were to bounce or move, its trajectory would be much longer than on Earth due to the decreased gravitational pull.

Understanding Martian gravity helps make accurate predictions about the motion of objects launched or moving in Mars's atmosphere. It also ensures the successful landing and operation of probes and rovers on the red planet.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects. In our Martian problem, kinematics helps break down the probe's movement into understandable components.

The horizontal and vertical components of motion are analyzed separately.

• The horizontal motion of the probe is constant because Martian gravity doesn’t affect it horizontally. We determined this component with the equation \( v_x = v \cos \theta \). This helped us find the horizontal velocity as \( 23.49 \, \mathrm{m/s} \).
• For the vertical motion, gravity influences it, and we used \( v_y = v \sin \theta \) to find the initial vertical speed, which is \( 8.55 \, \mathrm{m/s} \).

The understanding of time of flight and peak of motion are other critical points in kinematics. Using the motion equations, we calculated the total flight time as \(4.68\) seconds and time to peak as \(2.34\) seconds.

This break down helps grasp how objects move not just on Earth, but also on another planet like Mars.

Physics Problem Solving

Physics problem solving involves a systematic approach to understand and solve exercises related to real-world scenarios.

Here's how we solved the Martian probe problem:

• Break Down the Problem: We start by identifying known variables and what needs to be calculated, which were the initial speeds and the distance in this case.
• Apply Formulas: Use kinematic formulas to determine values like horizontal velocity, vertical velocity, and time of flight.
• Solve Equations: Substitute known values and solve equations step-by-step. This ensures that we find variables like time accurately.
• Verify the Solution: Rechecking is crucial to ensure every step follows logically from the previous and that the final calculation makes sense in the context.

Through such problem-solving strategies, students can enhance their analytical skills and develop a comprehensive understanding of physics concepts adapted to any planet's scenario.