Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 12a

Chapter 3, Problem 12a

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How much time is required for the football to reach the highest point of the trajectory?

Verified step by step guidance

Identify the vertical motion of the football. The initial upward velocity component is given as 12.0 m/s. The acceleration due to gravity, which acts downward, is approximately 9.8 m/s².

Understand that at the highest point of the trajectory, the vertical velocity of the football will be 0 m/s. This is because gravity will decelerate the upward motion until it stops momentarily before descending.

Use the kinematic equation for vertical motion to find the time to reach the highest point: \( v = u + at \), where \( v \) is the final vertical velocity (0 m/s at the highest point), \( u \) is the initial vertical velocity (12.0 m/s), \( a \) is the acceleration due to gravity (-9.8 m/s², negative because it opposes the upward motion), and \( t \) is the time.

Rearrange the equation to solve for time \( t \): \( t = \frac{v - u}{a} \). Substitute the known values into the equation: \( t = \frac{0 - 12.0}{-9.8} \).

Calculate the time \( t \) using the values substituted into the equation. This will give you the time required for the football to reach the highest point of its trajectory.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Projectile Motion

Projectile motion involves the movement of an object thrown into the air, subject to only the acceleration due to gravity. It can be analyzed by separating the motion into horizontal and vertical components, which are independent of each other. Understanding these components is crucial for solving problems related to the trajectory of projectiles.

Vertical Motion Under Gravity

Vertical motion under gravity refers to the motion of an object moving upwards or downwards under the influence of gravity. The upward velocity decreases until it reaches zero at the highest point, where the object momentarily stops before descending. The time to reach this point can be calculated using the initial vertical velocity and the acceleration due to gravity.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. These equations relate initial velocity, final velocity, acceleration, time, and displacement. For the vertical component of projectile motion, the equation v = u + at can be used to find the time taken to reach the highest point, where the final velocity is zero.

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Related Practice

Textbook Question

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)?

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Textbook Question

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How high is this point?

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A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How far has the football traveled horizontally during this time?

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