Answered

The table below shows the speed of a moving vehicle with respect to time.

\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Speed [tex]$(m/s)$[/tex] & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline
Time [tex]$(s)$[/tex] & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
\end{tabular}

(i) Find the acceleration of the vehicle.

(ii) Calculate the distance covered in 5 seconds.



Answer :

Let's solve the problem step-by-step.

### (i) Finding the acceleration of the vehicle:

Acceleration is defined as the change in velocity (speed) over time. We are given a list of speeds at different times, and we need to find the overall acceleration.

1. Identify the initial and final speeds:
- Initial speed, [tex]\( v_i \)[/tex]: [tex]\(0 \, \text{m/s}\)[/tex]
- Final speed, [tex]\( v_f \)[/tex]: [tex]\(10 \, \text{m/s}\)[/tex]

2. Identify the initial and final times:
- Initial time, [tex]\( t_i \)[/tex]: [tex]\(0 \, \text{s}\)[/tex]
- Final time, [tex]\( t_f \)[/tex]: [tex]\(5 \, \text{s}\)[/tex]

3. Calculate the change in velocity (Δv):
[tex]\[ \Delta v = v_f - v_i = 10 \, \text{m/s} - 0 \, \text{m/s} = 10 \, \text{m/s} \][/tex]

4. Calculate the change in time (Δt):
[tex]\[ \Delta t = t_f - t_i = 5 \, \text{s} - 0 \, \text{s} = 5 \, \text{s} \][/tex]

5. Use the formula for acceleration (a):
[tex]\[ a = \frac{\Delta v}{\Delta t} = \frac{10 \, \text{m/s}}{5 \, \text{s}} = 2 \, \text{m/s}^2 \][/tex]

So, the acceleration of the vehicle is [tex]\( 2 \, \text{m/s}^2 \)[/tex].

### (ii) Calculating the distance covered in 5 seconds:

To find the distance covered, we can use one of the kinematic equations for uniformly accelerated motion. We know:
- Initial speed, [tex]\( v_i \)[/tex]: [tex]\(0 \, \text{m/s}\)[/tex]
- Acceleration, [tex]\( a \)[/tex]: [tex]\(2 \, \text{m/s}^2\)[/tex]
- Time, [tex]\( t \)[/tex]: [tex]\(5 \, \text{s}\)[/tex]

The kinematic equation that relates these quantities is:
[tex]\[ \text{distance} = v_i \cdot t + \frac{1}{2} \cdot a \cdot t^2 \][/tex]

Plug in the known values:
[tex]\[ \text{distance} = 0 \cdot 5 \, \text{s} + \frac{1}{2} \cdot 2 \, \text{m/s}^2 \cdot (5 \, \text{s})^2 \][/tex]

Calculate each term:
[tex]\[ = 0 + \frac{1}{2} \cdot 2 \, \text{m/s}^2 \cdot 25 \, \text{s}^2 \][/tex]
[tex]\[ = \frac{1}{2} \cdot 2 \cdot 25 \][/tex]
[tex]\[ = 25 \, \text{m} \][/tex]

So, the distance covered by the vehicle in 5 seconds is [tex]\( 25 \, \text{meters} \)[/tex].

### Summary:
(i) The acceleration of the vehicle is [tex]\( 2 \, \text{m/s}^2 \)[/tex].

(ii) The distance covered in 5 seconds is [tex]\( 25 \, \text{meters} \)[/tex].