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4. Acceleration: Speeding Up or Slowing Down?

In the following questions, first estimate the answer before using an equation.

The velocity of a car that is moving east increases steadily from 20 m/s to 35 m/s in 5 s. Select east as the positive direction.

4.1 Draw a labeled velocity diagram to show the relationship between the change of velocity, initial velocity, and final velocity of the car. Also, draw a vector (arrow) to show the direction of acceleration.

4.2 Calculate the magnitude (size) of the car's average acceleration and its direction.

5. The car in Question 4 now slows down from 35 m/s east to 25 m/s in 4 s. Select east as the positive direction.

5.1 Draw a labeled vector diagram of the velocities in Question 5.

5.2 Calculate the magnitude (size) of the car's acceleration and give its direction.

5.3 Check your answer to Question 5.2 but now take west as the positive direction.



Answer :

GinnyAnswer
4.1 Velocity Diagram for Speeding Up:

First, let's visualize the motion of the car as it speeds up.

```
Initial Velocity (V_i1): 20 m/s (East)
Final Velocity (V_f1): 35 m/s (East)
Time Interval (t1): 5 s
```

To draw the velocity diagram:
- Start with a horizontal arrow pointing to the right (eastward) labeled "20 m/s".
- Next to it, draw another horizontal arrow pointing to the right but longer than the first one to indicate "35 m/s".
- The difference between the two velocities (increase in velocity) is represented by a third arrow pointing to the right, which shows the direction of the acceleration.

```
20 m/s Increase in Velocity (15 m/s)
|--------------------->|------------------------------>|
Initial Velocity Final Velocity

Acceleration Arrow pointing to the right (east)
```

4.2 Calculation of Average Acceleration (Speeding Up):

To calculate the magnitude of the car's average acceleration, we use the formula:
[tex]\[ \text{Acceleration} (a) = \frac{\text{Final Velocity} (V_f) - \text{Initial Velocity} (V_i)}{\text{Time} (t)} \][/tex]

Using the given values,
[tex]\[ V_i1 = 20 \, \text{m/s}, \quad V_f1 = 35 \, \text{m/s}, \quad t1 = 5 \, \text{s} \][/tex]

[tex]\[ a1 = \frac{35 \, \text{m/s} - 20 \, \text{m/s}}{5 \, \text{s}} \][/tex]

[tex]\[ a1 = \frac{15 \, \text{m/s}}{5 \, \text{s}} \][/tex]

[tex]\[ a1 = 3.0 \, \text{m/s}^2 \][/tex]

So, the magnitude of the car's average acceleration is [tex]\(3.0 \, \text{m/s}^2\)[/tex] directed eastward.

5.1 Velocity Diagram for Slowing Down:

Next, visualize the motion of the car as it slows down.

```
Initial Velocity (V_i2): 35 m/s (East)
Final Velocity (V_f2): 25 m/s (East)
Time Interval (t2): 4 s
```

To draw the velocity diagram:
- Start with a horizontal arrow pointing to the right (eastward) labeled "35 m/s".
- Next to it, draw another horizontal arrow pointing to the right but shorter than the first one to indicate "25 m/s".
- The difference between the two velocities (decrease in velocity) is represented by a third arrow pointing to the left, showing the direction of the acceleration.

```
35 m/s Decrease in Velocity (10 m/s)
|-------------------------------->|----------------->|
Initial Velocity Final Velocity

Acceleration Arrow pointing to the left (west)
```

5.2 Calculation of Average Acceleration (Slowing Down):

Using the formula for acceleration:
[tex]\[ a = \frac{V_f - V_i}{t} \][/tex]

Using the given values,
[tex]\[ V_i2 = 35 \, \text{m/s}, \quad V_f2 = 25 \, \text{m/s}, \quad t2 = 4 \, \text{s} \][/tex]

[tex]\[ a2 = \frac{25 \, \text{m/s} - 35 \, \text{m/s}}{4 \, \text{s}} \][/tex]

[tex]\[ a2 = \frac{-10 \, \text{m/s}}{4 \, \text{s}} \][/tex]

[tex]\[ a2 = -2.5 \, \text{m/s}^2 \][/tex]

So, the magnitude of the car's acceleration is [tex]\(2.5 \, \text{m/s}^2\)[/tex] directed westward (since the acceleration is negative when east is positive).

5.3 Acceleration with West as Positive Direction:

If we take west as the positive direction, a negative acceleration (deceleration) when moving east would now be a positive acceleration (as westward is positive).

[tex]\[ a2_{\text{west positive}} = -(-2.5 \, \text{m/s}^2) = 2.5 \, \text{m/s}^2 \][/tex]

So, with west as the positive direction, the magnitude of the car's acceleration remains [tex]\(2.5 \, \text{m/s}^2\)[/tex], but it is now positive.

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