Answer :
Let's break down the solution step-by-step for each part of the problem.
### 1. Bill's Acceleration
1. Initial Velocity of Bill: [tex]\( u = 7 \, \text{m/s} \)[/tex]
2. Final Velocity of Bill: [tex]\( v = 19 \, \text{m/s} \)[/tex]
3. Time taken by Bill: [tex]\( t = 5 \, \text{seconds} \)[/tex]
The formula for acceleration ([tex]\( a \)[/tex]) is:
[tex]\[ a = \frac{v - u}{t} \][/tex]
Substituting the values:
[tex]\[ a = \frac{19 - 7}{5} \][/tex]
[tex]\[ a = \frac{12}{5} \][/tex]
[tex]\[ a = 2.4 \, \text{m/s}^2 \][/tex]
So, Bill's acceleration is [tex]\( 2.4 \, \text{m/s}^2 \)[/tex].
### 2. Ben's Acceleration
1. Initial Velocity of Ben: [tex]\( u = 15 \, \text{m/s} \)[/tex]
2. Final Velocity of Ben: [tex]\( v = 21 \, \text{m/s} \)[/tex]
3. Time taken by Ben: [tex]\( t = 7 \, \text{seconds} \)[/tex]
Using the formula for acceleration ([tex]\( a \)[/tex]):
[tex]\[ a = \frac{v - u}{t} \][/tex]
Substituting the values:
[tex]\[ a = \frac{21 - 15}{7} \][/tex]
[tex]\[ a = \frac{6}{7} \][/tex]
[tex]\[ a \approx 0.8571 \, \text{m/s}^2 \][/tex]
So, Ben's acceleration is approximately [tex]\( 0.8571 \, \text{m/s}^2 \)[/tex].
### 3. Faster Velocity
To determine who is traveling at a faster velocity, we compare the final velocities of Bill and Ben.
- Bill's final velocity: [tex]\( 19 \, \text{m/s} \)[/tex]
- Ben's final velocity: [tex]\( 21 \, \text{m/s} \)[/tex]
Ben's final velocity is higher than Bill's.
Therefore, Ben is traveling at a faster velocity.
### Summary
1. Bill's acceleration is [tex]\( 2.4 \, \text{m/s}^2 \)[/tex].
2. Ben's acceleration is approximately [tex]\( 0.8571 \, \text{m/s}^2 \)[/tex].
3. Ben is traveling at a faster velocity (21 m/s).
### 1. Bill's Acceleration
1. Initial Velocity of Bill: [tex]\( u = 7 \, \text{m/s} \)[/tex]
2. Final Velocity of Bill: [tex]\( v = 19 \, \text{m/s} \)[/tex]
3. Time taken by Bill: [tex]\( t = 5 \, \text{seconds} \)[/tex]
The formula for acceleration ([tex]\( a \)[/tex]) is:
[tex]\[ a = \frac{v - u}{t} \][/tex]
Substituting the values:
[tex]\[ a = \frac{19 - 7}{5} \][/tex]
[tex]\[ a = \frac{12}{5} \][/tex]
[tex]\[ a = 2.4 \, \text{m/s}^2 \][/tex]
So, Bill's acceleration is [tex]\( 2.4 \, \text{m/s}^2 \)[/tex].
### 2. Ben's Acceleration
1. Initial Velocity of Ben: [tex]\( u = 15 \, \text{m/s} \)[/tex]
2. Final Velocity of Ben: [tex]\( v = 21 \, \text{m/s} \)[/tex]
3. Time taken by Ben: [tex]\( t = 7 \, \text{seconds} \)[/tex]
Using the formula for acceleration ([tex]\( a \)[/tex]):
[tex]\[ a = \frac{v - u}{t} \][/tex]
Substituting the values:
[tex]\[ a = \frac{21 - 15}{7} \][/tex]
[tex]\[ a = \frac{6}{7} \][/tex]
[tex]\[ a \approx 0.8571 \, \text{m/s}^2 \][/tex]
So, Ben's acceleration is approximately [tex]\( 0.8571 \, \text{m/s}^2 \)[/tex].
### 3. Faster Velocity
To determine who is traveling at a faster velocity, we compare the final velocities of Bill and Ben.
- Bill's final velocity: [tex]\( 19 \, \text{m/s} \)[/tex]
- Ben's final velocity: [tex]\( 21 \, \text{m/s} \)[/tex]
Ben's final velocity is higher than Bill's.
Therefore, Ben is traveling at a faster velocity.
### Summary
1. Bill's acceleration is [tex]\( 2.4 \, \text{m/s}^2 \)[/tex].
2. Ben's acceleration is approximately [tex]\( 0.8571 \, \text{m/s}^2 \)[/tex].
3. Ben is traveling at a faster velocity (21 m/s).
