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 maximum height the flea reached in the first $2.5$ ms.

Verified step by step guidance

1. Examine the second graph, which shows the speed of the flea in cm/s over time in milliseconds. The graph indicates that the flea reaches a maximum speed of approximately 150 cm/s at around 1.5 ms.
2. Convert the maximum speed from cm/s to m/s for consistency with standard units. Since 1 m = 100 cm, the speed in m/s is 150 cm/s divided by 100, which equals 1.5 m/s.
3. Use the kinematic equation for vertical motion to find the maximum height reached: \( h = v_i t + \frac{1}{2} a t^2 \), where \( v_i \) is the initial velocity (0 m/s, since the flea starts from rest), \( a \) is the acceleration, and \( t \) is the time.
4. Since the flea reaches its maximum speed at 1.5 ms, we can assume constant acceleration during this time. The acceleration \( a \) can be calculated using \( a = \frac{v_f - v_i}{t} \), where \( v_f \) is the final velocity (1.5 m/s) and \( t \) is 1.5 ms (converted to seconds: 0.0015 s).
5. Substitute the values into the kinematic equation to find the height reached in the first 2.5 ms. Note that the flea continues to move upwards after reaching maximum speed, but the acceleration is zero after 1.5 ms, so the height calculation is based on the initial acceleration phase.

Key Concepts

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this context, understanding the flea's jump requires analyzing its speed over time to determine how high it ascended during its jump.

Graph Interpretation

Interpreting graphs is crucial for extracting meaningful information from visual data representations. In this case, the speed-time graph shows how the flea's speed changes over time. By analyzing the graph, one can determine the maximum speed reached and the duration of the jump, which are essential for calculating the maximum height achieved.

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to gravitational forces. For the flea's jump, understanding the principles of projectile motion helps in calculating the maximum height reached based on its initial speed and the time of ascent. The vertical component of the jump is particularly important in this analysis.