Calculating Distance Traveled by a Body under Constant Acceleration

When a body starts from rest and experiences a constant acceleration, you can use the equation of motion to calculate the distance it travels. This article explains the steps and provides a detailed breakdown of how to solve such a problem.

Basic Equation for Distance Traveled under Constant Acceleration

The following equation can be used to find the distance traveled by a body starting from rest with constant acceleration:

[ d v_i t frac{1}{2} a t^2 ]

Where: d is the distance traveled vi is the initial velocity, which is 0 when starting from rest a is the acceleration t is the time

Given that the initial velocity vi 0, the equation simplifies to:

[ d frac{1}{2} a t^2 ]

Example Problem

Given the following parameters for a body moving with a constant acceleration:

Acceleration (a) 8 m/s2 Time (t) 5 seconds

Let's substitute these values into the equation:

[ d frac{1}{2} times 8 , text{m/s}^2 times (5 , text{s})^2 ]

Step 1: Square the time (t): [ (5 , text{s})^2 25 , text{s}^2 ]

Step 2: Calculate the product with half of the acceleration: [ d frac{1}{2} times 8 , text{m/s}^2 times 25 , text{s}^2 ]

Step 3: Perform the multiplication: [ d 4 , text{m/s}^2 times 25 , text{s}^2 100 , text{m} ]

The body travels a distance of 100 meters in 5 seconds.

Additional Examples to Enhance Understanding

Let's consider an additional example to further illustrate the concept:

Example 1: Using Final Velocity and Average Speed

Given:

Acceleration (a) 8 m/s2 Time (t) 5 seconds Initial velocity (vi) 0 m/s (starting from rest) Final velocity (vf) 240 m/s (after 4 seconds)

To find the distance traveled, follow these steps:

Step 1: Calculate the average speed: [ text{Average speed} frac{v_i v_f}{2} frac{0 240}{2} 120 , text{m/s} ]

Step 2: Use the average speed to find the distance traveled: [ text{Distance} text{Average speed} times text{Time} 120 , text{m/s} times 5 , text{s} 600 , text{m} ]

Example 2: Splitting the Time Interval

Suppose the body started from rest after 4 seconds, and you need to find the distance traveled in the 5th second:

Step 1: Calculate the speed after 4 seconds:

Initial speed (vi) 0 m/s Acceleration (a) 8 m/s2 Time (t) 4 seconds

Using the equation:

[ v v_i a t ]

[ v_f 0 8 times 4 32 , text{m/s} ]

Step 2: Calculate the distance traveled in 4 seconds:

[ s v_i t frac{1}{2} a t^2 ]

[ s 0 frac{1}{2} times 8 times 4^2 2 times 8 times 16 256 , text{m} ]

Step 3: Calculate the speed after 5 seconds:

[ v v_i a t ]

[ v_f 0 8 times 5 40 , text{m/s} ]

Step 4: Calculate the average speed in the 5th second:

[ text{Average speed in 5th second} frac{v_{4s} v_{5s}}{2} frac{32 40}{2} 36 , text{m/s} ]

Step 5: Calculate the distance traveled in the 5th second:

[ text{Distance in 5th second} text{Average speed} times text{Time} 36 , text{m/s} times 1 , text{s} 36 , text{m} ]

Conclusion

Understanding and applying the equations of motion, such as the distance equation for constant acceleration, can help solve complex physics problems. This article has provided a step-by-step guide on how to calculate the distance traveled by a body starting from rest with constant acceleration.

Keywords: constant acceleration, distance traveled, motion equations