Problem 18 - Physics for Scientists and Engineers with Modern Physics

Question

(II) \(\mathrm{A}\) 0.40-kg ball is thrown with a speed of 8.5 \(\mathrm{m} / \mathrm{s}\) at an upward angle of \(36^{\circ} .(a)\) What is its speed at its highest point, and \((b)\) how high does it go? (Use conservation of energy.)

Step-by-Step Solution

Answer

(a) Speed at highest point: 6.87 m/s; (b) Maximum height: 3.8 meters.

Step 1: Analyze the Problem

The ball is thrown with an initial speed of 8.5 m/s at an angle of 36 degrees. We need to determine two things using physics principles: the speed at the highest point and the maximum height achieved. To begin, decompose the velocity into horizontal and vertical components using trigonometry.

Step 2: Decompose the Initial Velocity

The initial velocity has a magnitude of 8.5 m/s and is directed at an angle of 36 degrees from the horizontal. The horizontal component is given by \(v_{x} = 8.5 \cdot \cos(36^{\circ})\) and the vertical component is \(v_{y} = 8.5 \cdot \sin(36^{\circ})\). Calculate these components using a calculator.

Step 3: Calculate the Speed at the Highest Point

At the highest point, the vertical component of velocity becomes zero due to gravity, while the horizontal component remains the same since there is no horizontal acceleration. Hence, the speed at the highest point is equal to the horizontal component, \(v_{x}\). Calculate \(v_{x}\) to find this speed.

Step 4: Apply Conservation of Energy to Find Maximum Height

Using the principle of conservation of energy, the initial kinetic energy plus the initial potential energy is equal to the final potential energy at the maximum height. Convert the kinetic energy initially associated with the vertical component of velocity into potential energy at maximum height: \[ \frac{1}{2} m v_{y}^{2} = mgh, \] where \( h \) is the maximum height and \( g \) is the acceleration due to gravity (9.8 m/s²). Solve for \( h \).

Step 5: Solve for Values

Substitute the known values: \( v_{x} = 8.5 \cdot \cos(36^{\circ}) \approx 6.87 \ m/s \) (speed at the highest point) and for height, solve \[ \frac{1}{2} (0.40) (8.5 \cdot \sin(36^{\circ}))^{2} = (0.40)(9.8)h \] to get \( h \). Calculate to get \( h \approx 3.8 \ m \).

Step 6: Conclusion

From our calculations, the speed of the ball at its highest point is approximately 6.87 m/s, and the highest point it reaches is approximately 3.8 meters.

Key Concepts

Projectile Motion

Projectile motion describes the trajectory of an object thrown into the air under the influence of gravity. It's one of the most fascinating elements of physics.
This type of motion occurs in two dimensions: horizontally and vertically.
The motion can be divided into two components:

• Horizontal motion, which moves at a constant speed because there's no horizontal acceleration.
• Vertical motion, affected by gravity, which accelerates downwards at a rate of 9.8 m/s².

By breaking the motion into these components, it is easier to analyze and solve related physics problems. Understanding the basics of projectile motion is crucial for solving problems related to objects like balls thrown into the air. In these scenarios, the initial angle and speed significantly influence how high and far the object will travel.

Kinematics

Kinematics is the branch of physics that studies motion without considering the forces that cause it. When analyzing projectile motion, kinematics offers key insights.
Key concepts of kinematics include:

• Velocity: The speed and direction of an object. In our example, the ball's initial velocity can be decomposed into horizontal and vertical components.
• Acceleration: For projectile motion, vertical acceleration is due to gravity, which is always directed downward.

Using kinematic equations, we can predict the position, speed, and acceleration of the projectile over time. By understanding these concepts, you can calculate how long a projectile is in the air and how high or far it will go.

Trigonometry

Trigonometry is the field of mathematics that deals with the relationships between angles and sides of triangles. It's crucial in decomposing the initial velocity of a projectile.
When you throw an object at an angle, trigonometry helps determine its horizontal and vertical velocity components.
To do this, you use:

• The cosine function to find the horizontal component: \(v_x = v \cdot \cos\theta\)
• The sine function to find the vertical component: \(v_y = v \cdot \sin\theta\)

Here, \(v\) is the initial velocity and \(\theta\) is the launch angle.
Decomposing the velocity is essential for analyzing projectile motion since it allows you to apply kinematic equations effectively. This process is important when solving for unknowns such as speed at a particular point or maximum height.