Question

A car starts from rest and accelerates at a uniform rate for 15 seconds reaching a constant speed of 30m/s

a) calculate the acceleration for the first 15 seconds
b) after T minutes, the total distance travelled is 45 kilometres
Find the value of T

Answer 1

The answer is 3minutes,32seconds

Explanation:

a)

`a = (v - u)/(t)`

`a = (30 - 0)/(15)`

a=30/15

`a = 2m {s}^( - 2)`

b) convert to km to m

45km=45000m

using

`s = ut + (1)/(2) a {t}^(2)`

`45000 = 0 + (1)/(2) * 2 * {t}^(2)`

`45000 = {t}^(2)`

take square root of both sides

`√(45000) = \sqrt{ {t}^(2) }`

t=212s

convert to minutes

3minutes,32seconds

Answer 2

a) Acceleration=
`\[ a = 2 \text{ m/s}^2 \]`

b) Value of T=12.125 minutes

To solve this problem, we can break it down into two parts: calculating the acceleration and then finding the total time taken to travel 45 kilometers.

Part a) Calculate the acceleration for the first 15 seconds

The formula for acceleration a when starting from rest (initial velocity u = 0 and reaching a final velocity v over time t is given by:

`\[ a = (v - u)/(t) \]`

Given:

- v = 30 m/s (final velocity)

- u = 0 m/s (initial velocity, because the car starts from rest)

- t = 15 s (time)

Let's calculate the acceleration:

`\[ a = \frac{30 \text{ m/s} - 0 \text{ m/s}}{15 \text{ s}} \]`

`\[ a = (30)/(15) \]`

`\[ a = 2 \text{ m/s}^2 \]`

So, the acceleration for the first 15 seconds is
`\( 2 \text{ m/s}^2 \).`

Part b) Find the value of T

The total distance travelled is the sum of the distance covered during the acceleration and the distance covered at constant speed.

For the acceleration phase:

The distance d1 covered under uniform acceleration can be found using the formula:

`\[ d_1 = u \cdot t + (1)/(2) a t^2 \]`

We know u = 0 ,
`\( a = 2 \text{ m/s}^2 \)`, and t = 15 s. Plugging these values in, we get:

`\[ d_1 = 0 \cdot 15 + (1)/(2) \cdot 2 \cdot 15^2 \]`

`\[ d_1 = (1)/(2) \cdot 2 \cdot 225 \]`

`\[ d_1 = 225 \text{ m} \]`

For the constant speed phase:

The total distance is 45 km, which is 45,000 m. We need to subtract the distance covered during acceleration to find the distance covered at constant speed d2 :

`\[ d_2 = 45,000 \text{ m} - 225 \text{ m} \]`

`\[ d_2 = 44,775 \text{ m} \]`

Now we find the time t2 it takes to cover d2 at a constant speed of
`\( v = 30 \text{ m/s} \):`

`\[ t_2 = (d_2)/(v) \]`

`\[ t_2 = \frac{44,775 \text{ m}}{30 \text{ m/s}} \]`

`\[ t_2 = 1492.5 \text{ s} \]`

To find the total time T in minutes, we add the acceleration time to t2 and convert to minutes:

`\[ T = \frac{(t_2 + 15 \text{ s})}{60} \]`

`\[ T = \frac{(1492.5 \text{ s} + 15 \text{ s})}{60} \]`

Let's calculate T .

The value of T , which is the total time taken to travel 45 kilometers, is approximately 25.125 minutes.