Question

1. the distance the car traveled in meters at 2 sec, 4 sec, 6 sec and 8 sec were recorded in the data table below. calculate the average velocity for the car after 4 seconds.

time (sec) | distance (m)
2 sec | 1 m
4 sec | 2 m
6 sec | 3 m
8 sec | 4 m
solution

Explanation:

Step1: Recall the formula for average velocity

The formula for average velocity \( v_{avg} \) is \( v_{avg}=\frac{\Delta d}{\Delta t} \), where \( \Delta d \) is the change in distance and \( \Delta t \) is the change in time. After 4 seconds, we consider the time interval from \( t_1 = 4 \) sec to \( t_2 \) (we can take the next time point, say \( t_2=6 \) sec, or we can also check the interval from \( t = 2 \) sec to \( t = 6 \) sec? Wait, no, the question is "after 4 seconds", so we should take the time interval starting from \( t = 4 \) sec. Let's check the data: at \( t = 4 \) sec, distance \( d_1=2 \) m; at \( t = 6 \) sec, distance \( d_2 = 3 \) m. So \( \Delta d=d_2 - d_1=3 - 2 = 1 \) m, \( \Delta t=t_2 - t_1=6 - 4 = 2 \) sec. Alternatively, we can also check the interval from \( t = 4 \) to \( t = 8 \): \( d_2 = 4 \) m, \( \Delta d=4 - 2 = 2 \) m, \( \Delta t=8 - 4 = 4 \) sec, but let's see the pattern. Wait, the data is linear: at 2 sec:1m, 4sec:2m, 6sec:3m, 8sec:4m. So the velocity is constant? Let's check the average velocity between 4 sec and 8 sec: \( \frac{4 - 2}{8 - 4}=\frac{2}{4}=0.5 \) m/s. Between 2 sec and 6 sec: \( \frac{3 - 1}{6 - 2}=\frac{2}{4}=0.5 \) m/s. Between 4 sec and 6 sec: \( \frac{3 - 2}{6 - 4}=\frac{1}{2}=0.5 \) m/s. So the average velocity after 4 seconds (using the interval from 4 sec to 8 sec or 4 sec to 6 sec) is 0.5 m/s. Let's do it properly. The average velocity is calculated as the change in distance over change in time. Let's take the time after 4 seconds, so we can use the time from \( t = 4 \) to \( t = 8 \) (a common interval). So \( \Delta d = 4 - 2 = 2 \) m, \( \Delta t = 8 - 4 = 4 \) sec. Then \( v_{avg}=\frac{\Delta d}{\Delta t}=\frac{2}{4}=0.5 \) m/s. Or from \( t = 4 \) to \( t = 6 \): \( \Delta d = 3 - 2 = 1 \) m, \( \Delta t = 6 - 4 = 2 \) sec, \( v_{avg}=\frac{1}{2}=0.5 \) m/s. Either way, the average velocity is 0.5 m/s.

Step1: Identify the formula for average velocity

The formula for average velocity is \( v_{avg}=\frac{\Delta d}{\Delta t} \), where \( \Delta d \) is the change in distance and \( \Delta t \) is the change in time.

Step2: Determine the time and distance values

We want the average velocity after 4 seconds. Let's take the time interval from \( t_1 = 4 \) sec to \( t_2 = 8 \) sec (we can also use \( t_2 = 6 \) sec, but using a larger interval is fine). At \( t_1 = 4 \) sec, the distance \( d_1 = 2 \) m. At \( t_2 = 8 \) sec, the distance \( d_2 = 4 \) m.

Step3: Calculate the change in distance and time

\( \Delta d = d_2 - d_1 = 4 - 2 = 2 \) m.
\( \Delta t = t_2 - t_1 = 8 - 4 = 4 \) sec.

Step4: Calculate the average velocity

Using the formula \( v_{avg}=\frac{\Delta d}{\Delta t} \), substitute the values:
\( v_{avg}=\frac{2}{4}=0.5 \) m/s.

Answer:

The average velocity of the car after 4 seconds is \( 0.5 \) m/s.