Time to Cover Distance with Uniform Acceleration: A Physics Problem Explained

In physics, it's often necessary to solve problems related to motion with uniform acceleration. One common question is: if a car starts from rest (with an initial velocity of 0 m/s) and covers a certain distance under uniform acceleration, how long will it take to cover that distance?

Let's delve into a specific problem: if a car starts from rest and covers a distance of 200 meters with a uniform acceleration of 4 m/s2, how long will it take to cover this distance?

Solving the Problem Using Kinematic Equations

To solve this, we can use the following kinematic equation:

s ut frac{1}{2}at2

Values Known:

s (distance covered) 200 meters u (initial velocity) 0 m/s (since the car starts from rest) a (acceleration) 4 m/s2 t (time in seconds) - which is what we need to find

Substituting these known values into the equation:

200 0 · t frac{1}{2} · 4 · t2

This simplifies to:

200 2t2

Now, solving for t2

t2 frac{200}{2} 100

Taking the square root of both sides gives:

t sqrt{100} 10 seconds

Thus, the time it takes for the car to cover 200 meters is 10 seconds.

Steps to Solve the Problem

1. List Known Values

Initial velocity (vo) 0 m/s Distance (s) 200 meters Acceleration (a) 4 m/s2 Final velocity (vf) - unknown initially

2. Use the Kinematic Equation to Solve for Time

Using the equation: s vot frac{1}{2}at2 Substitute: 200 0 · t frac{1}{2} · 4 · t2 Simplify: 200 2t2 Therefore: t2 100 So: t sqrt{100} 10 seconds

3. Verify Using Final Velocity

Find final velocity using: vf2 vo2 2as Substitute: vf2 02 2 · 4 · 200 Simplify: vf2 1600 Therefore: vf sqrt{1600} 40 m/s Use final velocity to find time: vf vo at Substitute: 40 0 4t Therefore: t frac{40}{4} 10 seconds

Conclusion

The time it takes for the car to cover 200 meters, given a uniform acceleration of 4 m/s2 and starting from rest, is 10 seconds. This is the same time that it would take for a public bus under similar circumstances. The solution confirms that the kinematic equations are consistent and reliable for solving such problems.

Additional Information

The formula for the time to cover distance from rest at acceleration is:

t sqrt{frac{2s}{a}}

Substituting the given values:

t sqrt{frac{2200}{4}} 10 seconds