Chapter 4: Problem 49
A softball is thrown from the origin of an \(x-y\) coordinate system with an initial speed of \(18 \mathrm{~m} / \mathrm{s}\) at an angle of \(35^{\circ}\) above the horizontal. (a) Find the \(x\) and \(y\) positions of the softball at the times \(t=0.50 \mathrm{~s}, 1.0 \mathrm{~s}, 1.5 \mathrm{~s}\), and \(2.0 \mathrm{~s}\). (b) Plot the results from part (a) on an \(x-y\) coordinate system, and sketch the parabolic curve that passes through them.
Short Answer
Step by step solution
Determine Initial Velocity Components
Calculate the X Position at Different Times
Calculate the Y Position at Different Times
Plot the Positions on Coordinate System
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
2D kinematics
- Position: Determining where the object is located at a specific time.
- Velocity: Understanding the speed and direction of the moving object.
- Acceleration: The rate of change of velocity, often influenced by gravity in projectile motion.
We can predict where the softball will land and how high it will travel over time by using these components.
Initial velocity components
To find the initial velocity components:
- Horizontal Component (\( v_{x,0} \)): This is calculated using the cosine of the launch angle. It represents the constant speed at which the object moves along the horizontal plane. In our exercise, this value is calculated as \( v_{x,0} = 18 \cdot \cos(35^{\circ}) \), resulting in approximately 14.74 m/s.
- Vertical Component (\( v_{y,0} \)): Found using the sine of the launch angle, this component determines how high and for how long the object will rise before gravity pulls it back down. For our softball, \( v_{y,0} = 18 \cdot \sin(35^{\circ}) \), equating to roughly 10.32 m/s.
Parabolic trajectory
As the softball in our problem is projected:
- The horizontal speed remains constant since there is no acceleration acting along the horizontal axis.
- The vertical motion is influenced by gravity, causing the softball to first rise to a peak and then descend, completing the parabola.
Coordinate system plotting
To effectively plot the trajectory of a softball:
- Identify the x and y positions at various time intervals, like the 0.5s, 1.0s, 1.5s, and 2.0s marks used in our exercise.
- Plot these points on a grid, where the x-axis shows horizontal distance and the y-axis shows vertical height.
This visual plot allows us to gauge the apex or peak height and predict points such as the landing location, crucial for real-world applications like sports or engineering.