An automobile driver puts on the brakes and decelerates from 30.0 m/s to zero in 15.0 s. What distance does the car travel?
To determine the distance traveled by the car, we can use the equation of motion:
\[d = v_i t + \frac{1}{2} a t^2\]
Where:
- \(d\) is the distance traveled
- \(v_i\) is the initial velocity
- \(t\) is the time taken
- \(a\) is the acceleration
In this case, the car is decelerating (negative acceleration), so we have:
\[v_i = 30.0 \, \text{m/s}\]
\(t = 15.0 \, \text{s}\)
\(a = -\frac{{v_i}}{{t}}\)
Now, let's solve step-by-step:
1. Calculate the acceleration:
\(a = -\frac{{30.0 \, \text{m/s}}}{{15.0 \, \text{s}}}\)
\(a = -2.0 \, \text{m/s}^2\)
2. Substitute the values into the equation of motion:
\[d = (30.0 \, \text{m/s})(15.0 \, \text{s}) + \frac{1}{2} (-2.0 \, \text{m/s}^2)(15.0 \, \text{s})^2\]
3. Simplify the equation:
\[d = 450 \, \text{m} - 0.5 \, (2.0 \, \text{m/s}^2)(225 \, \text{s}^2)\]
4. Compute the final result:
\[d = 450 \, \text{m} - 225 \, \text{m}\]
\[d = 225 \, \text{m}\]
Therefore, the car travels a distance of 225 meters.