A student drove to the university from her home and noted that the odometer reading of her car increased by $12.0 \mathrm{~km}$. The trip took 16.0 min. (For each answer, enter a number.)
(a) What was her average speed in $\mathrm{km} / \mathrm{h}$ ?
$\mathrm{km} / \mathrm{h}$
(b) If the straight-line distance from her home to the university is $10.3 \mathrm{~km}$ in a direction $25.0^{\circ}$ south of east, what was her average velocity in $\mathrm{km} / \mathrm{h}$ ? $\mathrm{km} / \mathrm{h}$
(c) If she returned home by the same path $7 \mathrm{~h} 30 \mathrm{~min}$ after she left, what were her average speed and velocity in $\mathrm{km} / \mathrm{h}$ for the entire trip?
average speed
$\mathrm{km} / \mathrm{h}$
average velocity
$\mathrm{km} / \mathrm{h}$

Step 1 ：Given that the distance travelled by the student is 12.0 km and the time taken is 16.0 min. We first convert the time from minutes to hours to match the units of the speed we want to calculate. This gives us \(\frac{16.0}{60} = 0.26666666666666666\) hours.

Step 2 ：We then calculate the average speed by dividing the total distance travelled by the total time taken. This gives us \(\frac{12.0}{0.26666666666666666} = 45.0\) km/h.

Step 3 ：The displacement (straight-line distance from start to end point) is given as 10.3 km. We calculate the average velocity by dividing the displacement by the total time taken. This gives us \(\frac{10.3}{0.26666666666666666} = 38.625\) km/h.

Step 4 ：For the entire trip, the total distance travelled is twice the distance from home to the university (since she returned by the same path), which is \(2 \times 12.0 = 24.0\) km. The total time taken is 7 hours 30 min after she left plus the time it took her to get to the university, which is \(7.5 + 0.26666666666666666 = 7.766666666666667\) hours.

Step 5 ：We calculate the average speed for the entire trip by dividing the total distance by the total time. This gives us \(\frac{24.0}{7.766666666666667} = 3.0901287553648067\) km/h.

Step 6 ：Since the student ended up at the same place she started, her total displacement is zero. Therefore, the average velocity for the entire trip is 0 km/h.

Step 7 ：\(\boxed{\text{Final Answer: }}\) (a) The average speed of the student was 45.0 km/h. (b) The average velocity of the student was 38.625 km/h. (c) The average speed for the entire trip was approximately 3.09 km/h and the average velocity was 0 km/h.