Motion and Motion Graphs: Question 7

Worked solution

Part (a): Ascent speed during the first section

Speed is the gradient of a distance–time (here, height–time) graph, $v = \dfrac{\Delta s}{\Delta t}$. During the first section, the height increases from $0\text{ m}$ to $300\text{ m}$ over $40\text{ s}$:

$v_1 = \frac{300\text{ m}}{40\text{ s}} = 7.5\text{ m/s}$

Part (b): Ascent speed during the second section

During the second section, the height increases from $300\text{ m}$ to $480\text{ m}$:

$\Delta s = 480\text{ m} - 300\text{ m} = 180\text{ m}, \qquad \Delta t = 100\text{ s} - 40\text{ s} = 60\text{ s}$

$v_2 = \frac{180\text{ m}}{60\text{ s}} = 3.0\text{ m/s}$

Comparing the two sections, $7.5\text{ m/s} > 3.0\text{ m/s}$, so the first section has the greater ascent speed. This is shown on the graph by the first section’s line being steeper than the second section’s line. A larger gradient means a greater speed, even though the balloon is still rising (not falling) in both sections.

Part (c): Interpreting the curved third section

The third section is a curve, not a straight line, so its gradient is not constant. The curve is steep immediately after $t = 100\text{ s}$ and becomes less steep as $t$ approaches $160\text{ s}$. Since the (downward) gradient of a height-time graph gives the descent speed, a gradient that is shrinking in magnitude means the balloon’s descent speed is decreasing throughout this section, it descends quickly at first and gradually slows down as it approaches the end of the section.

Part (d): Average descent speed over the third section

The area under a curve cannot be used to find a single speed here. Instead, the average speed over the whole section uses the total change in height and the total time taken, ignoring how the speed varies within that interval:

$\Delta s = 480\text{ m} - 120\text{ m} = 360\text{ m}, \qquad \Delta t = 160\text{ s} - 100\text{ s} = 60\text{ s}$

$v_{\text{avg}} = \frac{360\text{ m}}{60\text{ s}} = 6.0\text{ m/s}$

This value of $6.0\text{ m/s}$ is not the balloon’s actual descent speed at every instant. Because this section of the graph is a curve rather than a straight line, the descent speed is continuously changing (decreasing, as found in part (c)): it is greater than $6.0\text{ m/s}$ just after $t = 100\text{ s}$ and less than $6.0\text{ m/s}$ as $t$ approaches $160\text{ s}$. This is unlike the first and second sections, where the straight lines mean the calculated speed is the actual, constant speed for the whole section.

Part (e): Total distance travelled during the flight

The total distance travelled adds the size of the height change in every section, regardless of whether the balloon was rising or falling:

$\text{total distance} = 300\text{ m} + 180\text{ m} + 360\text{ m}$

$\text{total distance} = 840\text{ m}$

Final answers

• (a) Ascent speed in the first section $=$ $7.5\text{ m/s}$
• (b) Ascent speed in the second section $=$ $3.0\text{ m/s}$; the first section is faster, shown by its steeper line
• (c) The descent speed in the third section is decreasing, shown by the curve becoming less steep
• (d) Average descent speed over the third section $=$ $6.0\text{ m/s}$, which is only an average since the actual speed keeps changing
• (e) Total distance travelled during the whole flight $=$ $840\text{ m}$