2.5: Uniform Acceleration

Last updated
Save as PDF

Page ID: 2807

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

After jumping, this cliff diver undergoes effective free fall. Cliff diving is a sport in which athletes twist and flip on their way down. Even with the air resistance, these divers will accelerate the whole way down.

Uniform Acceleration

Acceleration that does not change in time is called uniform or constant acceleration. The velocity at the beginning of the time interval is called initial velocity, vi, and the velocity at the end of the time interval is called final velocity, vf. In a velocity versus time graph for uniform acceleration, the slope of the line is the acceleration. The equation that describes the curve is v_f=v_i+at.

Example 2.5.1

If an automobile with a velocity of 4.0 m/s accelerates at a rate of 4.0 m/s² for 2.5 s, what is the final velocity?

Solution

v_f=v_i+at=4.0 m/s+(4.0 m/s2)(2.5 s)=4.0 m/s+10. m/s=14 m/s

Example 2.5.2

If a cart slows from 22.0 m/s with an acceleration of -2.0 m/s², how long does it require to get to 4 m/s?

Solution

t=(v_f−v_i)/a=(−18 m/s−2.0 m/s)/2=9.0 s

Use the simulation below to further explore the uniform acceleration of a cliff diver. Observe the relationship between position, velocity, and acceleration as the diver falls under the influence of gravity. Begin by turning off the air resistance and see what happens when the diver falls as if he is in a vacuum. Then, if you are up for a challenge, turn the air resistance on and consider its effects on the diver.

Interactive Element

Summary

Acceleration that does not change in time is uniform, or constant, acceleration.
The equation relating initial velocity, final velocity, time, and acceleration is vf=vi+at.

Review

If an object has zero acceleration, does that mean it has zero velocity? Give an example.
If an object has zero velocity, does that mean it has zero acceleration? Give an example.
If the acceleration of a motorboat is 4.0 m/s², and the motorboat starts from rest, what is its velocity after 6.0 s?
The friction of the water on a boat produces an acceleration of -10. m/s². If the boat is traveling at 30. m/s and the motor is shut off, how long does it take the boat to slow down to 5.0 m/s?

Additional Resources

PLIX: Play, Learn, Interact, eXplore: Uniform Acceleration

Real World Application: Race to The Finish

Video:

Study Guide: Motion Study Guide