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Chapter 4: Problem 50

A baseball pitcher throws a pitch clocked at \(v_{x}=73.3 \mathrm{mi} / \mathrm{h}\). He throws horizontally. By what amount, \(d\), does the ball drop by the time it reaches home plate, \(L=60.0 \mathrm{ft}\) away? (a) First find a symbolic answer in terms of \(L, v_{x}\), and \(g\). (answer check available at lightandmatter.com) (b) Plug in and find a numerical answer. Express your answer in units of ft. (Note: \(1 \mathrm{ft}=12\) in, \(1 \mathrm{mi}=5280 \mathrm{ft}\), and \(1 \mathrm{in}=2.54 \mathrm{~cm}\) ) (answer check available at lightandmatter.com)

Short Answer

Expert verified
The ball drops approximately 4.98 ft by the time it reaches home plate.

Step by step solution

01

Determine Time of Flight (Symbolically)

To find the time it takes for the ball to reach home plate, we use the formula for time, which is distance divided by velocity. Here, the distance is represented by \(L\) and horizontal velocity by \(v_x\). Thus, the time \(t\) taken is \(t = \frac{L}{v_x}\).
02

Calculate Vertical Displacement (Symbolically)

The vertical motion of the baseball is governed by the equation for free fall. The vertical displacement \(d\) due to gravity can be expressed as \(d = \frac{1}{2} g t^2\), where \(g\) is the acceleration due to gravity (approximately \(32 \, \text{ft/s}^2\)). Substitute the time \(t\) from Step 1 to get \(d = \frac{1}{2} g \left(\frac{L}{v_x}\right)^2\).
03

Plug in Values to Find Numerical Time

Convert \(v_x\) from miles per hour to feet per second for consistency in units: \(v_x = 73.3 \, \text{mi/h}\; (\frac{5280 \, \text{ft}}{1 \, \text{mi}}) \; (\frac{1 \, \text{h}}{3600 \, \text{s}}) = 107.45 \, \text{ft/s}\). Use this to calculate the time \(t = \frac{60.0 \, \text{ft}}{107.45 \, \text{ft/s}} = 0.5582 \, \text{s}\).
04

Calculate Numerical Vertical Displacement

Now, substitute the calculated time onto the vertical displacement formula: \(d = \frac{1}{2} \times 32 \, \text{ft/s}^2 \times (0.5582 \, \text{s})^2 = 4.98 \, \text{ft}\).
05

Round the Numerical Answer

Round the final answer for vertical displacement to two decimal places: result is \(d = 4.98 \, \text{ft}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projectile Motion
Projectile motion refers to the motion of an object that is thrown or projected into the air and is subject to the force of gravity. This type of motion involves two components: horizontal and vertical. In the case of the baseball pitcher, the ball is thrown with a horizontal velocity but no initial vertical velocity.
  • The motion in the horizontal direction is uniform, meaning it maintains a constant velocity because no external horizontal forces act on it.
  • In the vertical direction, the motion is influenced by gravity, causing the object to accelerate downwards.
The key to solving a projectile motion problem is understanding how these two components interact. The horizontal and vertical motions are independent of one another and can be analyzed separately using different sets of equations, as depicted in the exercise.
Kinematics
Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. In this baseball throwing scenario, we use kinematics to determine how long it takes for the ball to reach its destination and how far it falls during this time.
  • Kinematic equations relate distance, velocity, acceleration, and time. For this problem, the horizontal motion is straightforward, as the ball travels a known distance, L, at a constant velocity, \(v_x\).
  • The equation \(t = \frac{L}{v_x}\) helps us find the time the ball takes to travel to home plate, indicating that time is proportional to distance and inversely proportional to velocity.
  • Vertical motion, controlled by gravity, uses the equation \(d = \frac{1}{2}g t^2\) to determine the displacement, showing that the fall is quadratic in time.
By deriving these values, we can anticipate the behavior of objects in motion through kinematics, gaining insights into how distance, velocity, and time relate to each other.
Free Fall
Free fall describes the motion of a body where gravity is the only force acting on it. While the horizontal component of the baseball's motion remains unaffected, its vertical motion can be understood as a free fall. Gravity pulls the ball downward, causing it to drop as it moves forward.
In this scenario, the vertical displacement can be calculated using the formula for an object in free fall: \(d = \frac{1}{2} g t^2\). Here:
  • \(g\) is the acceleration due to gravity (approximately \(32 \, \text{ft/s}^2\)).
  • \(t\) is the time the ball is in the air, calculated as \(t = \frac{L}{v_x}\) from the horizontal analysis.
By integrating these principles, we see that though the ball is thrown horizontally, it inevitably drops, due to gravitational force, by the time it reaches home plate. This understanding helps in anticipating how different variables, such as distance and angle, affect an object's trajectory when only gravity acts.

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Most popular questions from this chapter

Phnom Penh is \(470 \mathrm{~km}\) east and \(250 \mathrm{~km}\) south of Bangkok. Hanoi is \(60 \mathrm{~km}\) east and \(1030 \mathrm{~km}\) north of Phnom Penh. (a) Choose a coordinate system, and translate these data into \(\Delta x\) and \(\Delta y\) values with the proper plus and minus signs. (b) Find the components of the \(\Delta \mathbf{r}\) vector pointing from Bangkok to Hanoi.(answer check available at lightandmatter.com)

Many fish have an organ known as a swim bladder, an air-filled cavity whose main purpose is to control the fish's buoyancy an allow it to keep from rising or sinking without having to use its muscles. In some fish, however, the swim bladder (or a small extension of it) is linked to the ear and serves the additional purpose of amplifying sound waves. For a typical fish having such an anatomy, the bladder has a resonant frequency of \(300 \mathrm{~Hz}\), the bladder's \(Q\) is 3 , and the maximum amplification is about a factor of 100 in energy. Over what range of frequencies would the amplification be at least a factor of \(50 ?\)

A bird is initially flying horizontally east at \(21.1 \mathrm{~m} / \mathrm{s}\), but one second later it has changed direction so that it is flying horizontally and \(7^{\circ}\) north of east, at the same speed. What are the magnitude and direction of its acceleration vector during that one second time interval? (Assume its acceleration was roughly constant.) (answer check available at lightandmatter.com)

A car accelerates from rest. At low speeds, its acceleration is limited by static friction, so that if we press too hard on the gas, we will "burn rubber" (or, for many newer cars, a computerized traction-control system will override the gas pedal). At higher speeds, the limit on acceleration comes from the power of the engine, which puts a limit on how fast kinetic energy can be developed. (a) Show that if a force \(F\) is applied to an object moving at speed \(v\), the power required is given by \(P=v F\). (b) Find the speed \(v\) at which we cross over from the first regime described above to the second. At speeds higher than this, the engine does not have enough power to burn rubber. Express your result in terms of the car's power \(P\), its mass \(m\), the coefficient of static friction \(\mu_{s}\), and \(g\).(answer check available at lightandmatter.com) (c) Show that your answer to part b has units that make sense. (d) Show that the dependence of your answer on each of the four variables makes sense physically. (e) The 2010 Maserati Gran Turismo Convertible has a maximum power of \(3.23 \times 10^{5} \mathrm{~W}\) (433 horsepower) and a mass (including a 50 -kg driver) of \(2.03 \times 10^{3} \mathrm{~kg}\). (This power is the maximum the engine can supply at its optimum frequency of 7600 r.p.m. Presumably the automatic transmission is designed so a gear is available in which the engine will be running at very nearly this frequency when the car is at moving at \(v .\).) Rubber on asphalt has \(\mu_{s} \approx 0.9 .\) Find \(v\) for this car. Answer: \(18 \mathrm{~m} / \mathrm{s}\), or about 40 miles per hour. (f) Our analysis has neglected air friction, which can probably be approximated as a force proportional to \(v^{2}\). The existence of this force is the reason that the car has a maximum speed, which is 176 miles per hour. To get a feeling for how good an approximation it is to ignore air friction, find what fraction of the engine's maximum power is being used to overcome air resistance when the car is moving at the speed \(v\) found in part e.

A learjet traveling due east at \(300 \mathrm{mi} / \mathrm{hr}\) collides with a jumbo jet which was heading southwest at \(150 \mathrm{mi} / \mathrm{hr}\). The jumbo jet's mass is five times greater than that of the learjet. When they collide, the learjet sticks into the fuselage of the jumbo jet, and they fall to earth together. Their engines stop functioning immediately after the collision. On a map, what will be the direction from the location of the collision to the place where the wreckage hits the ground? (Give an angle.)(answer check available at lightandmatter.com)
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