How to Calculate the Time for a Car to Accelerate from 54km/h to 90km/h

A common problem in physics involves calculating the time required for a car to accelerate from one speed to another within a specific distance. This article will walk you through the process using the example where a car accelerates at a constant rate from 54 km/h to 90 km/h over a distance of 125 meters.

Converting Speeds to Meters per Second

Before we can solve for the acceleration and time, we need to convert the given speeds from kilometers per hour (km/h) to meters per second (m/s).

Conversion Calculations

Given:

Initial velocity, u 54 km/h 15 m/s Final velocity, v 90 km/h 25 m/s Distance, d 125 meters Time, t

Step 1: Compute the acceleration a

The formula for acceleration when a car travels a distance at a constant rate of acceleration is:

[a frac{v^2 - u^2}{2d}]

Substituting the given values:

[a frac{25^2 - 15^2}{2 times 125}]

Calculating the numerator:

252 - 152 625 - 225 400

Now, dividing by the distance 2 * 125:

400 / 250 1.6 m/s2

So, the acceleration, a 1.6 m/s2.

Calculating the Elapsed Time t

To find the time, we use the formula relating acceleration, initial velocity, and final velocity:

[v u at]

Arranging for t:

[t frac{v - u}{a}]

Substituting the values:

t frac{25 - 15}{1.6}

Calculating:

t frac{10}{1.6} 6.25 seconds

Therefore, it took 6.25 seconds to achieve a speed of 25 m/s, traveling 125 meters.

Alternative Approach

Another method to solve the problem is to use the average velocity approach. Here, we calculate the average velocity first and then use it to find the time taken.

Calculating Average Velocity

The change in velocity is:

90 km/h - 54 km/h 36 km/h

The average velocity, vavg, is the arithmetic mean of the initial and final velocities:

vavg frac{54 90}{2} 72 km/h

Converting Average Velocity to Meters per Second

Average velocity in m/s:

72 km/h * frac{1000}{3600} 20 m/s

Calculating the Time

Using the formula for time:

t frac{d}{vavg}

Substituting the values:

t frac{125}{20} 6.25 seconds

This confirms our previous calculation that it took 6.25 seconds to cover the distance of 125 meters under the given conditions.

Conclusion

Understanding and solving problems like this are crucial in physics, especially in engineering and car mechanics. By mastering these calculations, you can better understand how vehicles behave under different conditions and design safer and more efficient machinery.