Acceleration

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

Discussion

When the velocity of an object changes it is said to be accelerating or more formally acceleration is the rate of change of velocity with time.

In everyday English, the word acceleration is often used to describe a state of increasing speed. For many Americans, their only experience with acceleration comes from car ads. When a commercial shouts "zero to sixty in six point seven seconds" what they're saying here is that this particular car takes 6.7 s to reach a speed of 60 mph starting from a complete stop. This example illustrates acceleration as it is commonly understood, but acceleration in physics is much more than just increasing speed.

Any change in the velocity of an object results in an acceleration: increasing speed (what people usually mean when they say acceleration), decreasing speed (also called deceleration or retardation), or changing direction. Yes, that's right, a change in the direction of motion results in an acceleration even if the speed didn't change. That's because acceleration depends on the change in velocity and velocity is a vector quantity — one with both magnitude and direction. Thus, a falling apple accelerates, a car stopping at a traffic light accelerates, and an orbiting planet accelerates. Acceleration occurs anytime an object's speed increases, decreases, or changes direction.

Much like velocity, there are two kinds of acceleration: average and instantaneous. Average acceleration is measured over a "long" (that means measurable) time interval while instantaneous acceleration is measured over a "very small" (unbelievably short or infinitesimal) time interval. For each kind of acceleration, there's an equation …

a =  Δv
Δt
  average
acceleration		  
a =  lim Δv  =  dv  =  d2r
Δt→0 Δt dt dt2
  instantaneous
acceleration

For those of you familiar with calculus, check out the second equation, which states that acceleration is the first derivative of velocity with respect to time and the second derivative of displacement with respect to time. Or if you prefer, acceleration is the rate of change in velocity and also (since velocity is a change in displacement) the rate of change of the rate of change of displacement.

units

Calculating acceleration involves dividing velocity by time — or in terms of units, dividing meters per second [m/s] by second [s]. Dividing distance by time twice is the same as dividing distance by the square of time. Thus the SI unit of acceleration is the meter per second squared.

⎡
⎣		 m  =  m/s  =  m   1 ⎤
⎦
s2 s s s

Another frequently used unit is the acceleration due to gravity — g. Since we are all familiar with the effects of gravity on ourselves and the objects around us it makes for a convenient standard for comparing accelerations. Everything feels normal at 1 g, twice as heavy at 2 g, and weightless at 0 g. This unit has a very precise definition (g = 9.80665 m/s2) but for everyday use 9.8 m/s2 is sufficient.

The unit called acceleration due to gravity (represented by a roman g) is not the same as the natural phenomena called acceleration due to gravity (represented by an italic g). The former has a defined value whereas the latter has to be measured. (More on this later.)

Although the term "g force" is often used, the g is a measure of acceleration, not force. (More on this later.) Of particular concern to humans are the physiological effects of acceleration. To put things in perspective, all values are stated in g.

Here are some sample accelerations to end this section.

Automotive Acceleration (g)
event typical car sports car F-1 race car large truck
starting 0.3 - 0.5 > 0.9 1.7 < 0.2
braking 0.8 - 1.0 > 1.3 2 ~ 0.6
cornering 0.6 - 1.0 > 2.5 3 ??

 

Acceleration and the Human Body
a (g) event
2.9 sneeze
3.5 cough
3.6 crowd jostle
4.1 slap on back
8.1 hop off step
10.1 plop down in chair
60 chest acceleration limit during car crash at 48 km/h with airbag
70 - 100 crash that killed Diana, Princess of Wales, 1997
150 - 200 head acceleration limit during bicycle crash with helmet

Summary

Problems

practice

  1. A problem about a car (US version).
    1. A car is said to go "zero to sixty in six point seven seconds". What is its acceleration in m/s2?
    2. The driver can't release his foot from the gas pedal. (The gas pedal is also known as the accelerator. Coincidence? I think not.) How many additional seconds would it take for the driver to reach 80 mph (assuming the aceleration hasn't changed)?
    3. OK, enough with the English units. A car moving at 80 mph has a speed of 35.8 m/s. What acceleration would it have if it took 2.0 s to come to a complete stop?

    Solutions …

    1. Well first of all, we shouldn't be dealing with English units. They're truly difficult to work with, so let's convert them straight away and then do the old "plug and chug".
       
      v =  60 mile   1609 m   1 hour  = 26.8  m
      1 hour 1 mile 3600 s s
       
      a =  Δv  =  v − v0  =  26.8 m/s − 0 m/s  = 4.0  m
      Δt Δt 6.7 s s2
       
      Since the question asked for acceleration and acceleration is a vector quantity this answer is not complete. A proper answer must include a direction as well. This is quite easy to do. Since the car is starting from rest and moving forward, its acceleration must also be forward. The ultimate, complete answer to this problem is the car is accelerating at …
       
      4.0 m/s2 forward
       
    2. We should convert the final speed to SI units, use the fact that change equals rate times time, and then add that change to our velocity at the end of the previous problem. Algebra will do the rest for us.
       
      v =  80 mile   1609 m   1 hour  = 35.8  m
      1 hour 1 mile 3600 s s
       
      a =  Δv  =  v − v0
      Δt Δt
       
      Δt =  v − v0  =  35.8 m/s − 26.8 m/s  = 2.3 s
      a 4.0 m/s2
       
      Alternate solution …

      Once more with feeling. We don't need no stinkin' conversions with this method. The ratio of eighty to sixty is a simple one, namely 4/3. From our definition of acceleration, it should be apparent that time is directly proportional to change in velocity when acceleration is constant. Thus …
       
      Δv2  =  Δt2 ⇒ 80 mph  =  Δt2 ⇒ Δt2 = 8.9 s
      Δv1 Δt1 60 mph 6.7 s
       
      This is not the answer. It is the time elapsed from the moment when the car began to move. The question was about the additional time needed, so we should subtract the time required to go from zero to sixty. Thus …
       
      Δt = 8.9 s − 6.7 s = 2.2 s
       
      Ouch! Doesn't this show that the two different methods yield two different answers? Well, no, not really. What's happened is that rounding the results of one calculation and then using that in another has introduced an error -- a rounding error. The exact answer is somewhere in between 2.2 s and 2.3 s but it really doesn't matter in this sample problem. If we were really concerned with what the answer was we would keep track of every single digit all the way up to the final calculation. This is the best way to solve problems, but we're more concerned here with method than with solution so the difference between the two approaches is unimportant. They are, essentially, the same
    3. Quite simple. Let's do it.
       
      a =  Δv  =  v − v0  =  0 m/s − 35.8 m/s  = −17.9  m
      Δt Δt 2.0 s s2
       
      Nothing surprising there except the negative sign. When a vector quantity is negative what does it mean? There are several interpretations of this, but I think mine is the best. When a vector has a negative value, it means that it points in a direction opposite that of the positive vectors. In this problem, since the positive vectors are assumed to point forward (What other direction would a normal car drive?) the acceleration must be backward. Thus the complete answer to this problem is that the car's acceleration is …
       
      17.9 m/s2 backward
       
      Although it is common to assign deceleration a negative value, negative acceleration does not automatically imply deceleration. When dealing with vector quantities, any direction can be assumed positive …
       
      up, down, right, left, forward, backward, north, south, east, west
       
      and the corresponding opposite direction assumed negative …
       
      down, up, left, right, backward, forward, south, north, west, east.
       
      It won't matter which you chose as long as you are consistent throughout a problem. Don't learn any rules for assigning signs to particular directions and don't let anyone tell you that a certain direction must be positive or negative.
  2. I need a problem like the last one, but for the metric world.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. Which device(s) on a car can be used to control its acceleration?
  2. Describe a situation when an object has …
    1. zero velocity, but non-zero acceleration
    2. zero acceleration, but non-zero velocity

numerical

  1. At main engine cutoff (MECO), the Space Shuttle is at an altitude of 113 km (70 miles), traveling 7600 m/s (17,000 mph) relative to the earth. This occurs 7 minutes 40 seconds into the mission. Determine the magnitude of the average acceleration experienced by the shuttle astronauts from lift off to MECO.
  2. Most roller coasters are towed to the top of a large hill by means of a motor driven chain and released at the start of their run. This "chain link lift hill" technology is simple to design and quite reliable, but will never be able to accelerate the coaster faster than 1 g. In the quest to build new and ever more terrifying thrill rides, some designers have employed alternate acceleration methods. Two such roller coasters can be found at an amusement park just north of Richmond, Virginia.
    1. The Outer Limits: Flight of Fear is accelerated using linear induction motors (LIM), which generate a sequentially moving magnetic wave that propels the coaster like a surfer. A pair of LIMs is 85.3 m (280 foot) long and can accelerate the coaster to 24 m/s (54 mph) in 3.9 s. Determine the magnitude of the starting acceleration (in g) of the Flight of Fear.
    2. The HyperSonic XLC (Extreme Launch Coaster) is the world's first roller coaster to be launched using compressed air. Four, 150 kW (200 hp) compressed air motors accelerate the eight seat coaster from zero to 36 m/s (80 mph) in 1.8 s. Determine the magnitude of the starting acceleration (in g) of the Hypersonic XLC.
  3. When ejection seats were being developed, it was not known if a human could survive the intense acceleration needed to clear a jet fighter in an emergency. In 1954, US Air Force Colonel John Stapp was strapped into the seat of a rocket sled and blasted across the New Mexico desert at 282 m/s (632 mph) to examine the physiological effects of high speed ejection. The sled traveling at eight-tenths the speed of sound, a land speed record at that time, was then guided into a large trough of water, stopping it in a mere 1.4 s. Determine the magnitude of the average acceleration during the critical portion of this experiment. (Colonel Stapp subjected himself to several extreme acceleration experiments and survived all of them relatively unharmed.)
  4. Federal crash standards require that a passenger in a typical accident should not experience accelerations of 60 g for longer than 36 milliseconds. At what speed did the authors of this standard assume a typical accident would take place?
  5. During a typical accident, a properly designed bicycle helmet should keep acceleration of the head below 200 g for a cumulative duration of three milliseconds and 150 g for a cumulative duration of six milliseconds. At what speed did the authors of this standard assume a typical accident would take place?
  6. A distressed car is rolling backward, downhill at 3.0 m/s when its driver finally manages to get the engine started. What velocity will the car have 6.0 s later if it can accelerate at 3.0 m/s2?
  7. A baseball is pitched at 40 m/s (90 mph) in a Major League game. The batter hits the ball on a line drive straight toward the pitcher at 50 m/s (112 mph). Determine the magnitude of the acceleration of the ball if it was in contact with the bat for 1/30 s.
  8. What zero-to-sixty time is equivalent to an average acceleration of 1 g?

algebraic

  1. Prove that the difference of two adjacent squares is always an odd number (for example 9 − 4 = 5 or 16 − 9 = 7). What relation could this possibly have to one-dimensional motion with constant acceleration? (Galileo was probably the first person to make this connection.)

Resources

prev | up | next

Another quality webpage by

Glenn Elert		 eglobe logo home | contact

bent | chaos | eworld | facts | physics