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  4. Motion in one dimension

0.2 Motion in one dimension  (Page 7/16)

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Acceleration

Acceleration

Acceleration is the rate of change of velocity.

Acceleration (symbol a ) is the rate of change of velocity. It is a measure of how fast the velocity of an object changes in time. If we have a change in velocity ( Δ v ) over a time interval ( Δ t ), then the acceleration ( a ) is defined as:

acceleration ( in m · s - 2 ) = change in velocity ( in m · s - 1 ) change in time ( in s )
a = Δ v Δ t

Since velocity is a vector, acceleration is also a vector. Acceleration does not provide any information about a motion, but only about how the motion changes. It is not possible to tell how fast an object is moving or in which direction from the acceleration.

Like velocity, acceleration can be negative or positive. We see that when the sign of the acceleration and the velocity are the same, the object is speeding up. If both velocity and acceleration are positive, the object is speeding up in a positive direction. If both velocity and acceleration are negative, the object is speeding up in a negative direction. If velocity is positive and acceleration is negative, then the object is slowing down. Similarly, if the velocity is negative and the acceleration is positive the object is slowing down. This is illustrated in the following worked example.

A car accelerates uniformly from an initial velocity of 2 m · s - 1 to a final velocity of 10 m · s 1 in 8 seconds. It then slows down uniformly to a final velocity of 4 m · s - 1 in 6 seconds. Calculate the acceleration of the car during the first 8 seconds and during the last 6 seconds.

  1. Consider the motion of the car in two parts: the first 8 seconds and the last 6 seconds.

    For the first 8 seconds:

    v i = 2 m · s - 1 v f = 10 m · s - 1 t i = 0 s t f = 8 s

    For the last 6 seconds:

    v i = 10 m · s - 1 v f = 4 m · s - 1 t i = 8 s t f = 14 s

  2. For the first 8 seconds:

    a = Δ v Δ t = 10 m · s - 1 - 2 m · s - 1 8 s - 0 s = 1 m · s - 2

    For the next 6 seconds:

    a = Δ v Δ t = 4 m · s - 1 - 10 m · s - 1 14 s - 8 s = - 1 m · s - 2

    During the first 8 seconds the car had a positive acceleration. This means that its velocity increased. The velocity is positive so the car is speeding up. During the next 6 seconds the car had a negative acceleration. This means that its velocity decreased. The velocity is negative so the car is slowing down.

Acceleration does not tell us about the direction of the motion. Acceleration only tells us how the velocity changes.

Deceleration

Avoid the use of the word deceleration to refer to a negative acceleration. This word usually means slowing down and it is possible for an object to slow down with both a positive and negative acceleration, because the sign of the velocity of the object must also be taken into account to determine whether the body is slowing down or not.

Acceleration

  1. An athlete is accelerating uniformly from an initial velocity of 0 m · s - 1 to a final velocity of 4 m · s - 1 in 2 seconds. Calculate his acceleration. Let the direction that the athlete is running in be the positive direction.
  2. A bus accelerates uniformly from an initial velocity of 15 m · s - 1 to a final velocity of 7 m · s - 1 in 4 seconds. Calculate the acceleration of the bus. Let the direction of motion of the bus be the positive direction.
  3. An aeroplane accelerates uniformly from an initial velocity of 200 m · s - 1 to a velocity of 100 m · s - 1 in 10 seconds. It then accelerates uniformly to a final velocity of 240 m · s - 1 in 20 seconds. Let the direction of motion of the aeroplane be the positive direction.
    1. Calculate the acceleration of the aeroplane during the first 10 seconds of the motion.
    2. Calculate the acceleration of the aeroplane during the next 14 seconds of its motion.
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Source:  OpenStax, Physics - grade 10 [caps 2011]. OpenStax CNX. Jun 14, 2011 Download for free at http://cnx.org/content/col11298/1.3
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