Question

A car is traveling along a straight road at a velocity of +30.6 m/s when its engine cuts out. For the next 1.70 seconds, the car slows down, and its average acceleration is. For the next 4.56 seconds, the car slows down further, and its average acceleration is. The velocity of the car at the end of the 6.26-second period is +17.0 m/s. The ratio of the average acceleration values is = 1.55. Find the velocity of the car at the end of the initial 1.70-second interval.

Answer 1

The velocity of the car at the end of the initial 1.70-second interval is 25.6 m/s  

Let's evaluate every step

1. Step 1

$v_{i}_{1}$: initial velocity at step 1 = 30.6 m/s

$v_{f}_{1}$: final velocity at step 1

$t_{1}$: time at step 1 = 1.70 s

$a_{1}$: average acceleration at step 1

The average acceleration in this step is given by:

$a_{1} = \frac{v_{f_{1}} - v_{i_{1}}}{t_{1}}$   (1)      

                                                   

2. Step 2

$v_{i}_{2}$: initial velocity of step 2 = $v_{f}_{1}$

$v_{f}_{2}$: final velocity at step 2    

$t_{2}$: time at step 2 = 4.56 s

$a_{2}$: average acceleration at step 2

The average acceleration is the following:

$a_{2} = \frac{v_{f_{2}} - v_{i_{2}}}{t_{2}}$   (2)            

                 

3. Step 3          

This step is given by the sum of step 1 and step 2

$v_{f}$: final velocity after 6.26 s = 17.0 m/s = $v_{f}_{2}$

t = t₁ + t₂ = 1.70 s + 4.56 s = 6.26 s  

The ratio of the average acceleration is:

$\frac{a_{1}}{a_{2}} = 1.55$  (3)

By dividing equation (1) into equation (2) we have:

$\frac{a_{1}}{a_{2}} = \frac{\frac{v_{f_{1}} - v_{i_{1}}}{t_{1}}}{\frac{v_{f_{2}} - v_{i_{2}}}{t_{2}}}$

$\frac{a_{1}}{a_{2}} = \frac{t_{2}(v_{f_{1}} - v_{i_{1}})}{t_{1}(v_{f_{2}} - v_{f_{1}})}$                

By solving the above equation for $v_{f_{1}}$ we have:

$v_{f_{1}} = \frac{\frac{a_{1}}{a_{2}}v_{f_{2}}t_{1} + v_{i_{1}}t_{2}}{\frac{a_{1}}{a_{2}}t_{1} + t_{2}} = \frac{1.55*17.0 m/s*1.70 s + 30.6 m/s*4.56 s}{1.55*1.70 s + 4.56 s} = 25.6 m/s$

Therefore, the velocity of the car at the end of the initial 1.70-second interval is 25.6 m/s.  

 

You can see another example of average acceleration here: https://brainly.com/question/17050461?referrer=searchResults          

I hope it helps you!