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/s^{2}, 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_{f}^{2} – v_{o}^{2} = 2as = 2a (x_{f} – x_{o})
Sample question

A drag racer’s acceleration is 26.6 m/s^{2}, 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

To solve this problem, you need to relate speed, acceleration, and distance, so you start with this equation:
v_{f}^{2} – v_{o}^{2} = 2as = 2a (x_{f} – x_{o})

In this scenario, v_{o} is 0, which makes this equation simpler:
v_{f}^{2} = 2as

Solve for s:

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

Practice questions

A bullet is accelerated over a meterlong rifle barrel at an acceleration of 400,000 m/s^{2}. What is its final speed?

A dragster starts from rest and is accelerated at 5.0 m/s^{2}. What is its speed 5.0 x 10^{2} meters later?

A rocket is launched with an acceleration of 100.0 m/s^{2}. After 100.0 kilometers, what is its speed in meters per second?

A motorcycle is going 40.0 m/s and is accelerated at 6.00 m/s^{2}. What is its speed after 2.00 x 10^{2} meters?
Following are answers to the practice questions:

v_{f} = 900 m/s

Start with this equation:
v_{f}^{2} – v_{o}^{2} = 2as = 2a (x_{f} – x_{o})

v_{o} is 0, so that makes things easier. Plug in the numbers:

Take the square root:
v_{f} = 894 meters per second, which rounds to 900 with significant figures


2.v_{f} = 71 m/s

You want to find speed in terms of distance and acceleration, so use this equation:
v_{f}^{2} – v_{o}^{2} = 2as = 2a (x_{f} – x_{o})

Plug in the numbers:

Take the square root:
v_{f} = 70.7 meters per second, which rounds to 71 with significant figures


v_{f} = 4,472 m/s

You want to find the speed of the rocket, having been given distance and acceleration, so use this equation:
v_{f}^{2} – v_{o}^{2} = 2as = 2a (x_{f}_{ }– x_{o})

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

Take the square root to get the rocket’s speed:
v_{f }= 4,472 meters per second


v_{f} = 63.2 m/s

To determine the motorcycle’s final speed, use this equation:
v_{f}^{2} – v_{o}^{2 }= 2as

v_{0} = 40 meters per second, so plug in the numbers:

That means that v_{f}^{2} is
v_{f}^{2} = 4,000 (meters per second)^{2}

Take the square root to get v_{f}:
v_{f} = 63.2 meters per second
