Chapter 3: Problem 55
A baseball thrown at an angle of 60.0\(^{\circ}\) above the horizontal strikes a building 18.0 m away at a point 8.00 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the ball's initial velocity (the velocity with which the ball is thrown). (b) Find the magnitude and direction of the velocity of the ball just before it strikes the building.
Short Answer
Step by step solution
Setting up the problem
Determine horizontal and vertical components of motion
Analyze horizontal motion
Analyze vertical motion
Solve the equations simultaneously
Solve for initial velocity \(v_0\)
Obtain velocity before impact
Calculate magnitude and direction before impact
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Kinematics
In projectile motion, which is a kinematics problem, we analyze the movement of objects that are launched into the air. These objects move along a curved path due to their initial velocity and the force of gravity, which acts downward.
This path is called a trajectory, and it is typically parabolic in nature if we neglect air resistance.
- Displacement describes how far an object has moved from its starting point in a certain direction.
- Velocity involves speed in a specific direction, determining how fast the object travels.
- Acceleration refers to changes in velocity, such as those caused by gravity.
This helps explain the baseball's motion in our problem: it was launched at a point and moved upwards to an angle, finally hitting the wall a certain distance away. Breaking this into horizontal and vertical motions helps us make sense of the entire movement being analyzed.
Initial Velocity Calculation
To solve for initial velocity (\(v_0\)), we can use two motions:
- Horizontal motion, where no acceleration happens in the absence of air resistance.
- Vertical motion, which is affected by gravity.
The vertical motion formula, \(y = v_{0y}t - \frac{1}{2}gt^2\), incorporates acceleration due to gravity (\(g\)).
From these, two equations can be solved together by substituting the time of flight from the horizontal equation into the vertical equation. This enables us to find \(v_0\) using algebraic manipulation and solve for \(v_0 \approx13.9\text{ m/s}\), giving the magnitude of the ball's initial speed.
Angle of Projection
This angle plays a significant role in determining the trajectory's shape and how far the ball will land from its initial point.
The components of the velocity, horizontal (\(v_{0x}\)), and vertical (\(v_{0y}\)), are directly influenced by this angle and the initial velocity:
- Horizontal velocity: \(v_{0x} = v_0 \cos(60^{\circ})\)
- Vertical velocity: \(v_{0y} = v_0 \sin(60^{\circ})\)
- The horizontal component impacts how far the baseball travels horizontally.
- The vertical component influences how high the baseball flies and how long it stays in the air.