Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 52

Chapter 2, Problem 52

A cheetah spots a Thomson's gazelle, its preferred prey, and leaps into action, quickly accelerating to its top speed of 30 m/s, the highest of any land animal. However, a cheetah can maintain this extreme speed for only 15 s before having to let up. The cheetah is 170 m from the gazelle as it reaches top speed, and the gazelle sees the cheetah at just this instant. With negligible reaction time, the gazelle heads directly away from the cheetah, accelerating at 4.6 m/s² for 5.0 s, then running at constant speed. Does the gazelle escape? If so, by what distance is the gazelle in front when the cheetah gives up?

Verified step by step guidance

Step 1: Calculate the distance covered by the cheetah during its 15 seconds of running at its top speed. Use the formula for distance traveled at constant speed:

d = v \times t

, where

v

is the cheetah's top speed (30 m/s) and

t

is the time (15 s).

Step 2: Calculate the distance covered by the gazelle during its acceleration phase (5.0 s). Use the kinematic equation:

d = v

₀×t+12a×t2, where

v

₀ is the initial velocity (0 m/s),

a

is the acceleration (4.6 m/s²), and

t

is the time (5.0 s).

Step 3: Determine the gazelle's final velocity after the acceleration phase using the formula:

v = v

₀+a×t, where

v

₀ is the initial velocity (0 m/s),

a

is the acceleration (4.6 m/s²), and

t

is the time (5.0 s).

Step 4: Calculate the distance covered by the gazelle during the remaining 10 seconds (constant speed phase). Use the formula for distance traveled at constant speed:

d = v \times t

, where

v

is the gazelle's final velocity (calculated in Step 3) and

t

is the time (10 s).

Step 5: Compare the total distance covered by the gazelle (sum of distances from Steps 2 and 4) with the cheetah's total distance (Step 1 plus the initial 170 m gap). If the gazelle's total distance is greater, calculate the difference to determine how far ahead the gazelle is when the cheetah gives up.

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Key Concepts

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

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In this scenario, both the cheetah and the gazelle experience acceleration, which affects their speeds over time. The cheetah accelerates to its top speed quickly, while the gazelle accelerates at a constant rate before reaching a steady speed. Understanding how acceleration influences their motion is crucial for determining the outcome of the chase.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and time. In this problem, kinematic equations can be used to calculate the distances traveled by both the cheetah and the gazelle during their respective phases of motion, which is essential for determining whether the gazelle escapes.

Relative Motion

Relative motion refers to the calculation of the motion of an object as observed from another moving object. In this case, the motion of the cheetah is relative to the gazelle, and vice versa. Analyzing their speeds and accelerations relative to each other helps in understanding whether the gazelle can maintain a safe distance from the pursuing cheetah, which is key to solving the problem.

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04:27

Intro to Relative Motion (Relative Velocity)

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