Solving for Final Velocity

Solving for Final Velocity

We can derive another useful equation by manipulating the definition of acceleration.

\(a = \cfrac{\Delta v}{\Delta t}\)

Substituting the simplified notation for \(\Delta v\) and \(\Delta t\) gives us

\(a = \cfrac{v - v_0}{t} (\text{constant }a).\)

Solving for \(v\) yields:

Example on Calculating Final Velocity: An Airplane Slowing Down after Landing

An airplane lands with an initial velocity of 70.0 m/s and then decelerates at \(1.5\mathrm{\; m/s^2}\) for \(40.0\text{ s}\). What is its final velocity?

Strategy

Draw a sketch. We draw the acceleration vector in the direction opposite the velocity vector because the plane is decelerating.

Solution

1. Identify the knowns. \(v_0 = 70.0\text{ m/s},\) \(a = -1.50\mathrm{\; m/s^2},\) \(t = 40.0\text{ s}.\)

2. Identify the unknown. In this case, it is final velocity, \(v_{\text{f}}.\)

3. Determine which equation to use. We can calculate the final velocity using the equation \(v = v_0 + at.\)

4. Plug in the known values and solve.

\(v = v_0 + at = 70.0\text{ m/s} + \left ( -1.50\mathrm{ m/s^2} \right ) (40.0\text{ s}) = 10.0\text{ m/s}\)

Discussion

The final velocity is much less than the initial velocity, as desired when slowing down, but still positive. With jet engines, reverse thrust could be maintained long enough to stop the plane and start moving it backward. That would be indicated by a negative final velocity, which is not the case here.

In addition to being useful in problem solving, the equation \(v = v_0 + at\) gives us insight into the relationships among velocity, acceleration, and time. From it we can see, for example, that

• final velocity depends on how large the acceleration is and how long it lasts
• if the acceleration is zero, then the final velocity equals the initial velocity \(v = v_0,\) as expected (i.e., velocity is constant)
• if \(a\) is negative, then the final velocity is less than the initial velocity

(All of these observations fit our intuition, and it is always useful to examine basic equations in light of our intuition and experiences to check that they do indeed describe nature accurately.)

Making Connections: Real-World Connection

An intercontinental ballistic missile (ICBM) has a larger average acceleration than the Space Shuttle and achieves a greater velocity in the first minute or two of flight (actual ICBM burn times are classified—short-burn-time missiles are more difficult for an enemy to destroy). But the Space Shuttle obtains a greater final velocity, so that it can orbit the earth rather than come directly back down as an ICBM does. The Space Shuttle does this by accelerating for a longer time.