Ch 04: Kinematics in Two Dimensions

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 04: Kinematics in Two Dimensions

Problema 55b

Capitolo 4, Problema 55b

You're 6.0 m from one wall of the house seen in FIGURE P4.55. You want to toss a ball to your friend who is 6.0 m from the opposite wall. The throw and catch each occur 1.0 m above the ground. At what angle should you toss the ball?

Guida verificata passo dopo passo

Step 1: Identify the known values in the problem. The horizontal distance between you and your friend is the total distance between the walls, which is 6.0 m + 6.0 m = 12.0 m. The vertical displacement of the ball is zero because the throw and catch occur at the same height (1.0 m above the ground).

Step 2: Recall the kinematic equations for projectile motion. The horizontal motion is governed by the equation:

x = v

_xt, where

v

_x is the horizontal velocity component. The vertical motion is governed by:

y = v

_yt - 12gt², where

g

is the acceleration due to gravity.

Step 3: Break the initial velocity of the ball into components. The horizontal velocity is

v

_x = v₀⁢cos⁢θ, and the vertical velocity is

v

_y = v₀⁢sin⁢θ, where

v

₀ is the initial speed and

θ

is the angle of the throw.

Step 4: Use the condition that the vertical displacement is zero to find the time of flight. Set

y = 0

in the vertical motion equation and solve for

t

. This gives:

t = \frac{2}{v}

₀⁢sin⁢θg.

Step 5: Substitute the time of flight into the horizontal motion equation to find the angle. Use

x = v

_xt, and replace

v

_x and

t

with their respective expressions in terms of

v

₀ and

θ

. Solve for

θ

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Projectile Motion

Projectile motion refers to the motion of an object that is thrown into the air and is subject to the force of gravity. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravitational acceleration. Understanding the trajectory of the ball involves calculating the initial velocity, angle of projection, and the time of flight.

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Percorso guidato

04:44

Introduction to Projectile Motion

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In the context of projectile motion, these equations can be used to relate the initial velocity, angle of projection, time of flight, and displacement. For this problem, the horizontal and vertical components of the motion can be analyzed separately to determine the angle needed for the ball to reach the desired height and distance.

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Percorso guidato

08:25

Kinematics Equations

Angle of Projection

The angle of projection is the angle at which an object is thrown relative to the horizontal. It significantly affects the range and height of the projectile. For a given initial speed, there is an optimal angle (typically 45 degrees) that maximizes the distance traveled. However, in this scenario, the specific angle must be calculated to ensure the ball reaches a height of 1.0 m while covering the horizontal distance of 12.0 m.

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Critical Angle

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