Connecting Velocity, Acceleration, and Displacement in Physics Problems

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:

v_f² – v_o² = 2as = 2a (x_f – x_o)

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:

      v_f² – v_o² = 2as = 2a (x_f – x_o)

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

      v_f² = 2as

   3. Solve for s:

      

   4. Plug in the numbers:

      

      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. v_f = 900 m/s

   1. Start with this equation:

      v_f² – v_o² = 2as = 2a (x_f – x_o)

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

      

   3. Take the square root:

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

2. 2.v_f = 71 m/s

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

      v_f² – v_o² = 2as = 2a (x_f – x_o)

   2. Plug in the numbers:

      

   3. Take the square root:

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

3. v_f = 4,472 m/s

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

      v_f² – v_o² = 2as = 2a (x_f – x_o)

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

      

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

      v_f = 4,472 meters per second

4. v_f = 63.2 m/s

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

      v_f² – v_o² = 2as

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

      

   3. That means that v_f² is

      v_f² = 4,000 (meters per second)²

   4. Take the square root to get v_f:

      v_f = 63.2 meters per second