In addition to our position, velocity, and acceleration graphs, we can draw a motion map to describe the motion of the marble by showing which direction it went away from the origin. This is represented by the direction of the arrows (which, in this case, are pointed to the left). It also shows the increasing velocity as you read the motion map right to left, which is shown by the increasing arrow lengths. This means that the marble travels an even further distance as each second passes. Each second is represented by a dot.
a. To find the speed that it hits the water with, you are basically finding the velocity but ignoring the sign since, unlike velocity, speed does not take into account the direction of the motion. To mathematically find the velocity, you use the equation velocity equals acceleration times time plus initial velocity, or V = a*t + Vo.
However, we must find the acceleration first. This represents the slope of the vt graph, so by using the slope formula [slope = (y2 - y1) / (x2 - x1)] with points from the vt graph, you can find acceleration. Since the line is straight, you can choose any two points to plug in. By using (0s, 0m/s) and (6s, -58.8m/s), we can get the slope to equal the acceleration of the object. This relates to the motion of the object by telling us how much the velocity increases with every second that passes. In addition, we already know that time is 6s (given) and the initial velocity is 0m/s since it is starting from rest.
Therefore, the speed is 58.8m/s.
b. The height of the bridge is essentially the displacement (or change in position of the object in relation to its initial position) of the marble since we said the starting position of the marble was 0m. To get the displacement, you use the vt-graph of the motion and find the area between the line(s) that represent the motion and the t-axis. In this case, when we enclosed the area, it made a triangle, so to find the area of the triangle we use the simple geometric formula area equals 1/2 times base times height, or A = 1/2 * b * h. We use the length along the t-axis as the height and the v-value that corresponds to t = 6s since that will tell us the full length of the base.
However, since the height of something cannot be negative, we ignore the - sign and just say the height of the bridge is 176.4m.
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