Problem Analysis
The problem involves analyzing a velocity-time graph of a car traveling between two sets of traffic lights. The graph shows the velocity of the car as a function of time. We need to answer two questions based on this graph:
1.
Find the velocity of the car after 15 seconds.
2.
Estimate the acceleration of the car after 10 seconds.
---
Solution
#### Part (a): Velocity of the car after 15 seconds
To find the velocity of the car after 15 seconds, we need to look at the value of the velocity on the graph at \( t = 15 \) seconds.
- From the graph, at \( t = 15 \) seconds, the velocity is approximately
12 m/s.
Thus, the velocity of the car after 15 seconds is:
\[
\boxed{12}
\]
---
#### Part (b): Estimate the acceleration of the car after 10 seconds
Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, it is given by:
\[
a = \frac{\Delta v}{\Delta t}
\]
where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time.
To estimate the acceleration at \( t = 10 \) seconds, we need to determine the slope of the velocity-time graph at that point. The slope of the graph at any point gives the instantaneous acceleration.
1.
Identify the region around \( t = 10 \) seconds:
- At \( t = 10 \) seconds, the velocity is approximately
8 m/s.
- Choose a small interval around \( t = 10 \) seconds to estimate the slope. For example, we can use \( t = 9 \) seconds and \( t = 11 \) seconds.
2.
Find the velocities at \( t = 9 \) seconds and \( t = 11 \) seconds:
- At \( t = 9 \) seconds, the velocity is approximately
6 m/s.
- At \( t = 11 \) seconds, the velocity is approximately
10 m/s.
3.
Calculate the change in velocity (\( \Delta v \)) and the change in time (\( \Delta t \)):
- \( \Delta v = v_{11} - v_9 = 10 \, \text{m/s} - 6 \, \text{m/s} = 4 \, \text{m/s} \)
- \( \Delta t = t_{11} - t_9 = 11 \, \text{s} - 9 \, \text{s} = 2 \, \text{s} \)
4.
Estimate the acceleration:
\[
a = \frac{\Delta v}{\Delta t} = \frac{4 \, \text{m/s}}{2 \, \text{s}} = 2 \, \text{m/s}^2
\]
Thus, the estimated acceleration of the car after 10 seconds is:
\[
\boxed{2}
\]
---
Final Answers
\[
\boxed{12} \quad \text{(Part a)}
\]
\[
\boxed{2} \quad \text{(Part b)}
\]
Parent Tip: Review the logic above to help your child master the concept of velocity time graph worksheet.
