An inline skater traveling on a level road with a speed of 7.77 m/s is climbing a hill with a 15.08° angle of incline at a speed of 7.56 m/s. What is her average acceleration during the 120.0 s time interval?
The Correct Answer and Explanation is:
To find the average acceleration, we use the formula:aavg=vf−vita_{\text{avg}} = \frac{v_f – v_i}{t}aavg=tvf−vi
Step 1: Identify the known values
- Initial velocity vi=7.77 m/sv_i = 7.77 \, \text{m/s}vi=7.77m/s
- Final velocity vf=7.56 m/sv_f = 7.56 \, \text{m/s}vf=7.56m/s
- Time t=120.0 st = 120.0 \, \text{s}t=120.0s
Step 2: Plug into the formula
aavg=7.56 m/s−7.77 m/s120.0 s=−0.21 m/s120.0 s=−0.00175 m/s2a_{\text{avg}} = \frac{7.56 \, \text{m/s} – 7.77 \, \text{m/s}}{120.0 \, \text{s}} = \frac{-0.21 \, \text{m/s}}{120.0 \, \text{s}} = -0.00175 \, \text{m/s}^2aavg=120.0s7.56m/s−7.77m/s=120.0s−0.21m/s=−0.00175m/s2
Final Answer:
−0.00175 m/s2\boxed{-0.00175 \, \text{m/s}^2}−0.00175m/s2
Explanation:
The problem involves a skater moving initially on a level road and then ascending a hill. She starts at a speed of 7.77 meters per second and slows down to 7.56 meters per second over a time span of 120.0 seconds. The question asks for her average acceleration, which reflects the rate at which her speed changes over time.
Acceleration is defined as the change in velocity divided by the time taken. Since her speed is decreasing, the acceleration will be negative. A negative acceleration means she is decelerating or slowing down.
Using the formula a=(vf−vi)/ta = (v_f – v_i) / ta=(vf−vi)/t, we insert the initial and final speeds, along with the time interval. Subtracting the final speed from the initial speed gives a negative change in velocity, and dividing by the time results in a small negative number. This indicates a very gradual slowing down over a long time, which makes sense because she is only slowing slightly while going up a gentle incline.
The angle of the hill (15.08°) is mentioned, but for calculating average acceleration using speeds and time, it is not necessary. It would become relevant if we were calculating the forces involved or analyzing energy changes due to gravity.
The final answer, -0.00175 meters per second squared, tells us that her speed decreases by about 0.00175 meters per second every second. This is a very gentle deceleration, reflecting a slow and steady climb uphill.
