Connecting Velocity, Acceleration, and Displacement in Physics Problems

By Steven Holzner

You use the following equation to relate velocity, acceleration, and distance. Suppose you have a drag racer whose acceleration is 26.6 m/s2, and his final speed is 146.3 m/s. What is the total distance traveled? This scenario sets you up to use a handy motion equation:

vf2vo2 = 2as = 2a (xfxo)

Sample question

  1. A drag racer’s acceleration is 26.6 m/s2, and at the end of the race, his final speed is 146.3 m/s. What is the total distance the drag racer traveled?

    402 meters

    1. To solve this problem, you need to relate speed, acceleration, and distance, so you start with this equation:

      vf2vo2 = 2as = 2a (xfxo)

    2. In this scenario, vo is 0, which makes this equation simpler:

      vf2 = 2as

    3. Solve for s:

      image0.jpg

    4. Plug in the numbers:

      image1.jpg

      So the answer is 402 meters, about a quarter of a mile — standard for a drag racing track.

Practice questions

  1. A bullet is accelerated over a meter-long rifle barrel at an acceleration of 400,000 m/s2. What is its final speed?

  2. A dragster starts from rest and is accelerated at 5.0 m/s2. What is its speed 5.0 x 102 meters later?

  3. A rocket is launched with an acceleration of 100.0 m/s2. After 100.0 kilometers, what is its speed in meters per second?

  4. A motorcycle is going 40.0 m/s and is accelerated at 6.00 m/s2. What is its speed after 2.00 x 102 meters?

Following are answers to the practice questions:

  1. vf = 900 m/s

    1. Start with this equation:

      vf2vo2 = 2as = 2a (xfxo)

    2. vo is 0, so that makes things easier. Plug in the numbers:

      image2.jpg

    3. Take the square root:

      vf = 894 meters per second, which rounds to 900 with significant figures

  2. 2.vf = 71 m/s

    1. You want to find speed in terms of distance and acceleration, so use this equation:

      vf2vo2 = 2as = 2a (xfxo)

    2. Plug in the numbers:

      image3.jpg

    3. Take the square root:

      vf = 70.7 meters per second, which rounds to 71 with significant figures

  3. vf = 4,472 m/s

    1. You want to find the speed of the rocket, having been given distance and acceleration, so use this equation:

      vf2vo2 = 2as = 2a (xf xo)

    2. 100 kilometers is 100,000 meters, so plug in the numbers:

      image4.jpg

    3. Take the square root to get the rocket’s speed:

      vf = 4,472 meters per second

  4. vf = 63.2 m/s

    1. To determine the motorcycle’s final speed, use this equation:

      vf2vo2 = 2as

    2. v0 = 40 meters per second, so plug in the numbers:

      image5.jpg

    3. That means that vf2 is

      vf2 = 4,000 (meters per second)2

    4. Take the square root to get vf:

      vf = 63.2 meters per second