91Ó°ÊÓ

Acceleration is a measure of how quickly an object's velocity changes. When we talk about acceleration, it's important to remember that it's a vector quantity. This means it has both magnitude and direction. For the cheetah problem, we're interested in how the cheetah speeds up from rest to its final velocity.To find acceleration, we use the formula:\[ a = \frac{v_f - v_0}{t} \]where:- \( v_f \) is the final velocity, \( 27 \, \text{m/s} \) for the cheetah.- \( v_0 \) is the initial velocity, \( 0 \, \text{m/s} \) because the cheetah starts from rest.- \( t \) is the time over which this change occurs, \( 4.0 \, \text{s} \).Plugging in these values, we find that the cheetah's acceleration is \( 6.75 \, \text{m/s}^2 \). This tells us how fast the cheetah increases its speed every second.

Kinetic energy is the energy an object possesses due to its motion. It's given by the formula:\[ KE = \frac{1}{2} m v^2 \]where:- \( m \) is the mass of the object.- \( v \) is the velocity of the object.For our cheetah, with a mass of \( 110 \, \text{kg} \) and a final velocity of \( 27 \, \text{m/s} \), we calculated:\[ KE = \frac{1}{2} \times 110 \times 27^2 = 40095 \, \text{J} \]This represents the kinetic energy when the cheetah reaches its top speed. Energy is a key factor when looking at the cheetah's performance, as it reflects how much work has been done to accelerate it to this speed.

In physics, being able to convert units correctly is crucial. In this problem, we needed to convert the cheetah's power from watts to horsepower.Remember that power, which is the rate of doing work, is initially given in watts \((W)\), the standard unit in the International System of Units. However, horsepower, another common unit, is often used in contexts involving engines or powerful animals like our cheetah.To convert between these units, use the conversion factor:- \( 1 \, \text{horsepower} = 746 \, \text{watts} \)Thus, if the cheetah generates \( 10023.75 \, \text{W} \), we follow:\[ \text{Horsepower} = \frac{10023.75}{746} \approx 13.44 \, \text{hp} \]This shows how much power the cheetah would need in terms of engines and other power sources, providing a tangible comparison.

The work-energy principle is pivotal in understanding how energy changes in a system. It states that the work done on an object is equal to the change in its kinetic energy.For the cheetah, this means as it accelerates, the work done by its muscles converts into kinetic energy. This can be calculated by:\[ W = \Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_0^2 \]Given the initial and final velocity are \( 0 \, \text{m/s} \) and \( 27 \, \text{m/s} \) respectively, and the mass is \( 110 \, \text{kg} \), the work done becomes:\[ W = 40095 \, \text{J} \]This direct link between work and energy helps explain how the cheetah can achieve such rapid acceleration, transforming stored energy into the energy of motion effortlessly.