High-speed motion pictures ($3500$ frames/second) of a jumping, $210–μg$ flea yielded the data used to plot the graph in Fig. E$2.54$. (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Find the flea's acceleration at $0.5$ ms, $1.0$ ms, and $1.5$ ms.

Verified step by step guidance

1. To find the flea's acceleration at specific times, we need to determine the slope of the velocity-time graph at those times. Acceleration is the rate of change of velocity with respect to time.
2. Examine the graph provided. The graph shows velocity (in cm/s) on the y-axis and time (in ms) on the x-axis. We need to find the slope of the graph at 0.5 ms, 1.0 ms, and 1.5 ms.
3. At 0.5 ms, identify the velocity from the graph. The slope at this point is the change in velocity divided by the change in time. Use the formula: \( a = \frac{\Delta v}{\Delta t} \). Calculate the slope using the velocities at points slightly before and after 0.5 ms.
4. Repeat the process for 1.0 ms. Find the velocity at 1.0 ms from the graph, and calculate the slope using the velocities at points slightly before and after 1.0 ms.
5. Finally, repeat the process for 1.5 ms. Find the velocity at 1.5 ms from the graph, and calculate the slope using the velocities at points slightly before and after 1.5 ms. This will give you the acceleration at each of these times.

Key Concepts

Acceleration

Acceleration is the rate of change of velocity over time. It is a vector quantity, meaning it has both magnitude and direction. In the context of the flea's jump, acceleration can be determined by analyzing the change in speed over specific time intervals, which can be derived from the speed-time graph provided.

Speed-Time Graph

A speed-time graph visually represents an object's speed as a function of time. The slope of the graph indicates acceleration; a steeper slope means greater acceleration. In this case, the graph shows how the flea's speed increases rapidly at first and then levels off, indicating a change in acceleration over time.

Instantaneous Speed

Instantaneous speed refers to the speed of an object at a specific moment in time. It can be determined from the speed-time graph by reading the speed value at a given time. For the flea's jump, finding the instantaneous speed at 0.5 ms, 1.0 ms, and 1.5 ms will help in calculating the corresponding accelerations.