91Ó°ÊÓ

When we talk about motion, particularly in the context of physics, we often encounter the concept of uniformly accelerated motion. This is a type of motion where an object's velocity changes at a constant rate over time. More simply, it's when something speeds up or slows down steadily. In our electric cart example, the cart starts from rest and reaches a speed of 26.82 m/s in 15 seconds.
This means the cart experiences uniform acceleration, calculated using the formula:

• Acceleration (\(a\)) = \(\frac{\text{final speed} - \text{initial speed}}{\text{time}}\)

Here, the initial speed is 0 (since it starts from rest), the final speed is 26.82 m/s, and the time taken is 15 seconds. By substituting these values, the acceleration comes out to be 1.788 m/s².

This steady increase in speed allows us to use specific equations to predict its motion, including how far it travels over a given period.

Understanding how to calculate distance under uniformly accelerated motion is crucial in this exercise. The electric cart covers a certain distance while it is accelerating, which we calculate using the equation for uniformly accelerated motion:

• Formula: Distance (\(S\)) = \(ut + \frac{1}{2}at^2\)

In this equation, \(u\) stands for the initial velocity, \(a\) for acceleration, and \(t\) for time. Since our cart starts from rest, the initial velocity \(u\) is 0. Substituting acceleration (\(a = 1.788\) m/s²) obtained earlier and time (\(t = 15\) seconds) into the formula gives us the distance traveled, which equates to approximately 200.82 meters.
The accuracy of such calculations helps in accurately predicting how far an object will move under certain conditions.

Converting speeds from one unit to another is often necessary to perform accurate calculations and comparisons. In our scenario, the speed of the electric cart is initially in miles per hour (mph), but to be applicable in equations more common in physics, we need it in meters per second (m/s).

• Conversion Factor: 1 mph = approximately 0.447 m/s

Using this conversion factor, we transform the cart's maximum speed from 60 mph to m/s:

• Speed in m/s = 60 \(\times 0.447 = 26.82\)

This conversion allows us to use the speed in equations for distance and acceleration calculating purposes, making it easy to compare with other measurements, like the athletic performance in meters per second, allowing for accurate analysis and decisions.

The comparison between the electric cart and Usain Bolt's sprint performance helps us understand not only physics but also the real-world implications of these calculations.
Usain Bolt runs 100 meters in 9.69 seconds, giving him an impressive average speed of:

• Speed = \(\frac{100}{9.69} = 10.32\) m/s

During the same 9.69 seconds, using the principles of uniformly accelerated motion, we calculate the distance the cart travels, considering its acceleration:

• Distance = \(\frac{1}{2} \times 1.788 \times (9.69)^2 = 83.255\) meters

Here, we see Bolt clearly outpaces the cart by approximately 16.74 meters in a 100-meter race.
Such comparisons exemplify the usefulness of applying physics not only in theoretical scenarios but in understanding and analyzing real-world performances like those of elite athletes.