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Constant acceleration is a fundamental concept in kinematics. It occurs when an object's velocity changes at a consistent rate over time. Imagine a car speeding up at a steady pace when you press the accelerator pedal all the way to the floor. This predictable increase in speed is a prime example of constant acceleration.

For the cheetah in our exercise, starting from rest and reaching a velocity of 25.0 meters per second in 6 seconds demonstrates constant acceleration. To determine this rate of change in velocity, we use the formula:

• \( v = u + at \)

Where:

• \( v \) is the final velocity (25.0 m/s)
• \( u \) is the initial velocity (0 m/s since it starts from rest)
• \( a \) is the acceleration
• \( t \) is the time duration (6.00 s)

By rearranging the formula to solve for acceleration \( a \), we find that the cheetah has a constant acceleration of 4.17 m/s².

Calculating how far an object travels under constant acceleration involves another basic kinematic equation. This is useful when you need to know the distance covered within a certain time frame while an object is accelerating. In our scenario, we need to find the distance the cheetah covers in the first 3 seconds. Here’s the formula we use:

• \( s = ut + \frac{1}{2}at^2 \)

In this equation:

• \( s \) is the distance traveled
• \( u \) is the initial velocity (0 m/s in this case since we started from rest)
• \( a \) is the constant acceleration (4.17 m/s²)
• \( t \) represents the time duration in question (3.00 seconds here)

Substituting these values into the formula, we calculate:

• \( s = 0 + \frac{1}{2} \times 4.17 \times (3.00)^2 = 18.8 \, \mathrm{m} \)

Naturally, this means that the cheetah covers 18.8 meters in the first 3 seconds under constant acceleration.

Velocity, another fundamental concept in kinematics, describes how fast an object is moving in a specific direction. It is a vector quantity, meaning it has both magnitude and direction. Unlike speed, which is scalar and only considers how fast an object is moving, velocity must specify the direction of movement.

In our cheetah exercise, the velocity at the end of 6 seconds is 25.0 m/s. This final velocity plays a crucial role in finding acceleration when the cheetah starts from rest. Understanding how velocity changes over time when an object is experiencing constant acceleration helps us predict motion paths and distances covered accurately.

By visualizing velocity as a straight-line increase in our graph of time versus velocity, we can see the nature of constant acceleration. As all velocity starts at 0 and steadily increases to 25.0 m/s over a period of 6 seconds, it's clear that the cheetah gains velocity in equal increments each second, thanks to its constant acceleration.