Question

section 2 acceleration continued
how can you calculate acceleration?
for an object moving in a straight line, acceleration occurs only because of changes in speed. therefore, you can calculate the object’s acceleration if you know its speed at two different times. you can use the equation below to calculate acceleration:
acceleration = (final speed - initial speed) / time
( a = \frac{delta v}{t} )
in this equation, the symbol “delta” (( delta )) means “change in.” you calculate acceleration by dividing the change in speed by the time in which the change occurred.
if the acceleration is small, the velocity is changing slowly. for example, a person can accelerate at about ( 2 , \text{m/s}^2 ). if the acceleration is large, the velocity is changing more quickly. a sports car can accelerate at about ( 7.2 , \text{m/s}^2 ).
calculating acceleration from velocity
let’s look at an example. a cyclist slows along a straight line from ( 5.5 , \text{m/s} ) to ( 1.0 , \text{m/s} ) in ( 3.0 , \text{s} ). what is the average acceleration of the cyclist?
| step 1: list the given and unknown values. | given:
initial speed, ( v_i = 5.5 , \text{m/s} )
final speed, ( v_f = 1.0 , \text{m/s} )
time, ( t = 3.0 , \text{s} ) | unknown:
acceleration, ( a ) |

step 2: write the equation.

( a = \frac{delta v}{t} = \frac{v_f - v_i}{t} )

| step 3: insert the known values and solve for the unknown value. | ( a = \frac{1.0 , \text{m/s} - 5.5 , \text{m/s}}{3.0 , \text{s}} )
( a = \frac{-4.5 , \text{m/s}}{3.0 , \text{s}} )
( a = -1.5 , \text{m/s}^2 ) |

critical thinking

1. infer can you use the equation to the left to calculate centripetal acceleration? explain your answer.

math skills

1. calculate a turtle swimming in a straight line toward shore has a speed of ( 0.50 , \text{m/s} ). after ( 4.0 , \text{s} ), its speed is ( 0.80 , \text{m/s} ). what is its average acceleration? show your work.

Explanation:

Problem 6: Calculate the average acceleration of the turtle

Step 1: Identify given values

Initial speed \( v_i = 0.50 \, \text{m/s} \), final speed \( v_f = 0.80 \, \text{m/s} \), time \( t = 4.0 \, \text{s} \)

Step 2: Write the acceleration formula

The formula for acceleration is \( a=\frac{\Delta v}{t}=\frac{v_f - v_i}{t} \)

Step 3: Substitute values into the formula

Substitute \( v_f = 0.80 \, \text{m/s} \), \( v_i = 0.50 \, \text{m/s} \), and \( t = 4.0 \, \text{s} \) into the formula:
\( a=\frac{0.80 \, \text{m/s}- 0.50 \, \text{m/s}}{4.0 \, \text{s}} \)

Step 4: Calculate the numerator and then the acceleration

First, calculate the numerator: \( 0.80 \, \text{m/s}- 0.50 \, \text{m/s}=0.30 \, \text{m/s} \)
Then, divide by the time: \( a = \frac{0.30 \, \text{m/s}}{4.0 \, \text{s}} = 0.075 \, \text{m/s}^2 \)

Answer:

The average acceleration of the turtle is \( 0.075 \, \text{m/s}^2 \)