Problem 15 - University Physics with Modern Physics

Question

Inside a starship at rest on the earth, a ball rolls off the top of a horizontal table and lands a distance \(D\) from the foot of the table. This starship now lands on the unexplored Planet \(X\). The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on earth and finds that it lands a distance 2.76\(D\) from the foot of the table. What is the acceleration due to gravity on Planet \(X\)?

Step-by-Step Solution

Answer

The acceleration due to gravity on Planet X is approximately 1.29 m/s².

Step 1: Calculate Time of Flight on Earth

When the ball rolls off the table on Earth, it lands a distance \(D\) from the table. Let \(h\) be the height of the table and \(v\) be the initial horizontal velocity of the ball. The time \(t\) it takes to fall height \(h\) is given by the equation \(h = \frac{1}{2}gt^2\). Here, \(g\) is the acceleration due to gravity on Earth, which is approximately \(9.81 \, \text{m/s}^2\). Solving for \(t\) gives \(t = \sqrt{\frac{2h}{g}}\).

Step 2: Calculate Horizontal Velocity on Earth

The horizontal distance travelled by the ball is \(D = vt\). Substituting the expression for \(t\) from the previous step gives \(D = v \sqrt{\frac{2h}{g}}\). We can rearrange to find the horizontal velocity \(v = \frac{D}{\sqrt{\frac{2h}{g}}}\).

Step 3: Calculate Time of Flight on Planet X

On Planet \(X\), the ball travels a horizontal distance of \(2.76D\). The time of flight \(t'\) is expressed using the height and the unknown acceleration due to gravity \(g'\) as \(h = \frac{1}{2}g' (t')^2\). Solving for \(t'\) gives \(t' = \sqrt{\frac{2h}{g'}}\).

Step 4: Relate Horizontal Distances on Planet X

The horizontal distance on Planet \(X\) is \(2.76D = v \cdot t'\). Substitute \(t' = \sqrt{\frac{2h}{g'}}\) into the equation to get \(2.76D = v \sqrt{\frac{2h}{g'}}\).

Step 5: Solve for Gravity on Planet X

From our previous substitutions, we have two equations of motion: \(v = \frac{D}{\sqrt{\frac{2h}{g}}}\) and \(2.76D = v \sqrt{\frac{2h}{g'}}\). Substituting \(v\) from the Earth equation into the Planet \(X\) equation yields \(2.76D = \frac{D}{\sqrt{\frac{2h}{g}}} \cdot \sqrt{\frac{2h}{g'}}\). Cancelling terms and rearranging gives \(g' = \frac{g}{2.76^2}\).

Step 6: Calculate Value of Gravity on Planet X

Substitute the known value of gravity on Earth \(g = 9.81 \, \text{m/s}^2\) into our expression for \(g'\). We find \(g' = \frac{9.81}{2.76^2} \approx 1.29 \, \text{m/s}^2\).

Key Concepts

Projectile Motion

Projectile motion is a fascinating aspect of physics that deals with objects that are thrown or projected into the air. These objects follow a curved path under the influence of gravity. When you let a ball roll off a table inside a starship, for instance, it covers horizontal ground before landing:

- **Horizontal Motion**: The horizontal component is governed by the initial speed with which it was rolled. It doesn't change (ignoring air resistance), meaning constant velocity in that direction.
- **Vertical Motion**: Simultaneously, the ball falls vertically under the force of gravity. This causes it to accelerate downward.

The combination of these two independent motions—horizontal at constant velocity and vertical accelerating due to gravity—forms the projectile's path. This parabolic trajectory is what we refer to as projectile motion. Understanding this dual-motion aspect helps in analyzing how far and fast the object travels horizontally before touching the ground.

Acceleration Due to Gravity

Acceleration due to gravity is a crucial force that influences how objects move in projectile motion. It's the rate at which an object's velocity increases as it falls towards a celestial body, like Earth or Planet X. The standard gravity on Earth is about 9.81 m/s².

When comparing different environments, like Earth and another planet, gravity can vary. For instance, Planet X in our example accelerates objects slower than Earth, with a gravity value of approximately 1.29 m/s². This can lead to significant differences in how far a projectile travels:

• Stronger gravity (like Earth's) results in a shorter time in the air and a shorter horizontal travel distance.
• Weaker gravity (Planet X) means the object stays in the air longer, traveling a farther horizontal distance.

Understanding the role of gravity helps us predict and calculate projectile motion outcomes on different worlds.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It uses variables like velocity, acceleration, time, and displacement to describe how objects move.

To solve the problem given in our exercise, kinematic equations are essential. They help relate the height it falls, time in the air, and distances traveled. Key concepts include:

• **Time of Flight**: Calculated based on vertical motion using the formula \( h = \frac{1}{2}gt^2 \), where \( h \) is the height of the table.
• **Horizontal Motion**: Using distance \( D = vt \), where \( v \) is the horizontal velocity.
• **Gravity's Impact**: Different gravitational accelerations on Earth and Planet X need adjustments in the calculations of flight time and distances.

By understanding these kinematic relationships, we can effectively predict how objects behave when projected, whether on Earth or on a mysterious new planet.