Answer :
Let's solve this problem step-by-step using the five equations of uniformly accelerated motion.
Given:
- Initial velocity, [tex]\( v_i = 10 \, \text{m/s} \)[/tex]
- Final velocity, [tex]\( v_f = 18 \, \text{m/s} \)[/tex]
- Time, [tex]\( t = 3 \, \text{s} \)[/tex]
We need to find:
1. The acceleration [tex]\( a \)[/tex]
2. The initial velocity [tex]\( v_i \)[/tex]
3. The displacement [tex]\( d \)[/tex]
4. The final velocity squared [tex]\( v_f^2 \)[/tex]
5. The displacement using initial and final velocities
### 1. Finding Acceleration ([tex]\( a \)[/tex])
Using the first equation of uniformly accelerated motion:
[tex]\[ v_f = v_i + at \][/tex]
Rearranging to solve for acceleration [tex]\( a \)[/tex]:
[tex]\[ a = \frac{v_f - v_i}{t} \][/tex]
Substituting the given values:
[tex]\[ a = \frac{18 \, \text{m/s} - 10 \, \text{m/s}}{3 \, \text{s}} \][/tex]
[tex]\[ a = \frac{8 \, \text{m/s}}{3 \, \text{s}} \][/tex]
[tex]\[ a \approx 2.67 \, \text{m/s}^2 \][/tex]
### 2. Verification of Initial Velocity ([tex]\( v_i \)[/tex])
We are given the initial velocity ([tex]\( v_i = 10 \, \text{m/s} \)[/tex]).
### 3. Finding Displacement ([tex]\( d \)[/tex])
Using the third equation of uniformly accelerated motion:
[tex]\[ d = v_i t + \frac{1}{2} a t^2 \][/tex]
Substituting the given values:
[tex]\[ d = 10 \, \text{m/s} \times 3 \, \text{s} + \frac{1}{2} \times 2.67 \, \text{m/s}^2 \times (3 \, \text{s})^2 \][/tex]
[tex]\[ d = 30 \, \text{m} + \frac{1}{2} \times 2.67 \, \text{m/s}^2 \times 9 \, \text{s}^2 \][/tex]
[tex]\[ d = 30 \, \text{m} + 12.015 \, \text{m} \][/tex]
[tex]\[ d \approx 42.015 \, \text{m} \][/tex]
### 4. Finding Final Velocity Squared ([tex]\( v_f^2 \)[/tex])
Using the fourth equation of uniformly accelerated motion:
[tex]\[ v_f^2 = v_i^2 + 2ad \][/tex]
Substituting the given values:
[tex]\[ v_f^2 = (10 \, \text{m/s})^2 + 2 \times 2.67 \, \text{m/s}^2 \times 42.015 \, \text{m} \][/tex]
[tex]\[ v_f^2 = 100 \, \text{m}^2/\text{s}^2 + 2 \times 2.67 \times 42.015 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ v_f^2 = 100 \, \text{m}^2/\text{s}^2 + 224.971 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ v_f^2 \approx 324.971 \, \text{m}^2/\text{s}^2 \][/tex]
### 5. Finding Displacement Using Initial and Final Velocities
Using the fifth equation of uniformly accelerated motion:
[tex]\[ d = \frac{1}{2} (v_i + v_f) t \][/tex]
Substituting the given values:
[tex]\[ d = \frac{1}{2} (10 \, \text{m/s} + 18 \, \text{m/s}) \times 3 \, \text{s} \][/tex]
[tex]\[ d = \frac{1}{2} \times 28 \, \text{m/s} \times 3 \, \text{s} \][/tex]
[tex]\[ d = 14 \, \text{m/s} \times 3 \, \text{s} \][/tex]
[tex]\[ d = 42 \, \text{m} \][/tex]
So, following the five equations of uniformly accelerated motion, we calculated:
1. Acceleration ([tex]\( a \)[/tex]) [tex]\(\approx 2.67 \, \text{m/s}^2\)[/tex]
2. Initial Velocity ([tex]\( v_i \)[/tex]) = [tex]\(10 \, \text{m/s} \)[/tex]
3. Displacement ([tex]\( d \)[/tex]) [tex]\(\approx 42.015 \, \text{m} \)[/tex]
4. Final Velocity Squared ([tex]\( v_f^2 \)[/tex]) [tex]\(\approx 324.971 \, \text{m}^2/\text{s}^2 \)[/tex]
5. Displacement using Initial and Final Velocities: [tex]\( d = 42 \, \text{m} \)[/tex]
Given:
- Initial velocity, [tex]\( v_i = 10 \, \text{m/s} \)[/tex]
- Final velocity, [tex]\( v_f = 18 \, \text{m/s} \)[/tex]
- Time, [tex]\( t = 3 \, \text{s} \)[/tex]
We need to find:
1. The acceleration [tex]\( a \)[/tex]
2. The initial velocity [tex]\( v_i \)[/tex]
3. The displacement [tex]\( d \)[/tex]
4. The final velocity squared [tex]\( v_f^2 \)[/tex]
5. The displacement using initial and final velocities
### 1. Finding Acceleration ([tex]\( a \)[/tex])
Using the first equation of uniformly accelerated motion:
[tex]\[ v_f = v_i + at \][/tex]
Rearranging to solve for acceleration [tex]\( a \)[/tex]:
[tex]\[ a = \frac{v_f - v_i}{t} \][/tex]
Substituting the given values:
[tex]\[ a = \frac{18 \, \text{m/s} - 10 \, \text{m/s}}{3 \, \text{s}} \][/tex]
[tex]\[ a = \frac{8 \, \text{m/s}}{3 \, \text{s}} \][/tex]
[tex]\[ a \approx 2.67 \, \text{m/s}^2 \][/tex]
### 2. Verification of Initial Velocity ([tex]\( v_i \)[/tex])
We are given the initial velocity ([tex]\( v_i = 10 \, \text{m/s} \)[/tex]).
### 3. Finding Displacement ([tex]\( d \)[/tex])
Using the third equation of uniformly accelerated motion:
[tex]\[ d = v_i t + \frac{1}{2} a t^2 \][/tex]
Substituting the given values:
[tex]\[ d = 10 \, \text{m/s} \times 3 \, \text{s} + \frac{1}{2} \times 2.67 \, \text{m/s}^2 \times (3 \, \text{s})^2 \][/tex]
[tex]\[ d = 30 \, \text{m} + \frac{1}{2} \times 2.67 \, \text{m/s}^2 \times 9 \, \text{s}^2 \][/tex]
[tex]\[ d = 30 \, \text{m} + 12.015 \, \text{m} \][/tex]
[tex]\[ d \approx 42.015 \, \text{m} \][/tex]
### 4. Finding Final Velocity Squared ([tex]\( v_f^2 \)[/tex])
Using the fourth equation of uniformly accelerated motion:
[tex]\[ v_f^2 = v_i^2 + 2ad \][/tex]
Substituting the given values:
[tex]\[ v_f^2 = (10 \, \text{m/s})^2 + 2 \times 2.67 \, \text{m/s}^2 \times 42.015 \, \text{m} \][/tex]
[tex]\[ v_f^2 = 100 \, \text{m}^2/\text{s}^2 + 2 \times 2.67 \times 42.015 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ v_f^2 = 100 \, \text{m}^2/\text{s}^2 + 224.971 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ v_f^2 \approx 324.971 \, \text{m}^2/\text{s}^2 \][/tex]
### 5. Finding Displacement Using Initial and Final Velocities
Using the fifth equation of uniformly accelerated motion:
[tex]\[ d = \frac{1}{2} (v_i + v_f) t \][/tex]
Substituting the given values:
[tex]\[ d = \frac{1}{2} (10 \, \text{m/s} + 18 \, \text{m/s}) \times 3 \, \text{s} \][/tex]
[tex]\[ d = \frac{1}{2} \times 28 \, \text{m/s} \times 3 \, \text{s} \][/tex]
[tex]\[ d = 14 \, \text{m/s} \times 3 \, \text{s} \][/tex]
[tex]\[ d = 42 \, \text{m} \][/tex]
So, following the five equations of uniformly accelerated motion, we calculated:
1. Acceleration ([tex]\( a \)[/tex]) [tex]\(\approx 2.67 \, \text{m/s}^2\)[/tex]
2. Initial Velocity ([tex]\( v_i \)[/tex]) = [tex]\(10 \, \text{m/s} \)[/tex]
3. Displacement ([tex]\( d \)[/tex]) [tex]\(\approx 42.015 \, \text{m} \)[/tex]
4. Final Velocity Squared ([tex]\( v_f^2 \)[/tex]) [tex]\(\approx 324.971 \, \text{m}^2/\text{s}^2 \)[/tex]
5. Displacement using Initial and Final Velocities: [tex]\( d = 42 \, \text{m} \)[/tex]
