Video Transcript
A car is driven along a straight road. It starts at rest. It is first driven forward and then slows down and reverses and finally ends up back at its starting point. The displacement of the car along the road over time is shown by the red line on the graph. The blue line is a tangent to the red line at 𝑡 equals 16 seconds. Points 𝐴 and 𝐵 mark where the blue line crosses grid lines. What is the speed of the car at 𝑡 equals 16 seconds? Give your answer to two decimal places.
The graph shows time 𝑡 along the horizontal axis and displacement 𝑑 on the vertical axis. If we follow the path of the car given by the red line, we can see it’s starting at rest and then driving forwards, slowing down, and then reversing and finally returning to its starting point. We need to find the speed of the car at 𝑡 equals 16 seconds. So let’s first find 𝑡 equals 16 seconds on our horizontal axis. And if we move upwards from the axis to find the red line, we can find that we’re looking for the speed just after the car has begun to reverse.
So let’s first recall how to find speed from a displacement–time graph. Speed is the magnitude or size of the slope of a displacement–time graph, where by magnitude we just mean the positive value because a negative speed makes no physical sense. To find the slope of a curve, we need a tangent, which is a straight line that touches the curve and has the same slope as the curve at the point where they touch. So we need a tangent to this point where 𝑡 equals 16 seconds. And the question tells us that the blue line is a tangent to the red line at 𝑡 equals 16 seconds.
So we need to find the slope of the blue line. And recall that the slope is the vertical difference divided by the horizontal difference between any two points on a line. Now, it’s easiest to use two points where the line crosses grid lines. And the question gives us these two points 𝐴 and 𝐵, where the blue line crosses grid lines. So let’s find the coordinates first of point A, which is at five, 20, and then of point 𝐵, which is at 26, zero.
So the slope is the vertical difference between these two points, which is 20 minus zero, divided by the horizontal difference, which is five minus 26. So on the top, we have 20 minus zero, which is 20. And on the bottom, we have five minus 26, which is equal to negative 21. We then take 20 divided by minus 21, which is equal to minus 0.952 et cetera. Now, we want to turn this into a speed for which we need to find the magnitude of the slope. This just means the positive value. So the positive value of minus 0.952 is just 0.952.
The question asks for the speed to two decimal places. So this is 0.95 with the two rounding down. And finally, we need to add some units. So we take the units of the vertical axis, which are meters, and we divide by the units of the horizontal axis, which are seconds. And this gives us our final answer that the speed of the car at 𝑡 equals 16 seconds is 0.95 meters per second.
