Question

(a) If a flea can jump straight up to a height of 0.440 m, what is its initial speed as it leaves the ground? (b) How long is it in the air?

Short answer

(a) Initial speed is 2.94 m/s. (b) Time in air is 0.60 seconds.

Step-by-step solution

Understand the Problem

We need to find the initial speed of the flea as it leaves the ground (part a) and the total time it is in the air (part b). The maximum height reached by the flea is 0.440 m.

Apply the Kinematic Equation for (a)

To find the initial speed, we use the kinematic equation: \[ v^2 = u^2 + 2as \]where \( v \) is the final velocity (0 m/s at the maximum height), \( u \) is the initial speed, \( a \) is the acceleration due to gravity (-9.81 m/s², negative because it acts downward), and \( s \) is the maximum height (0.440 m). We solve for \( u \):\[ 0 = u^2 - 2 \times 9.81 \times 0.440 \]

Solve for Initial Speed

Rearrange the equation to find \( u \): \[ u^2 = 2 \times 9.81 \times 0.440 \]\[ u = \sqrt{2 \times 9.81 \times 0.440} \]Calculate to find \( u \, \approx 2.94 \text{ m/s} \).

Use Time of Flight Formula for (b)

The time to reach the maximum height can be found with \( v = u + at \), simplified as \( 0 = u - 9.81t \), where \( t \) is the time to reach the top.\[ t = \frac{u}{9.81} = \frac{2.94}{9.81} \approx 0.30 \text{ seconds} \]

Calculate Total Time in Air

The total time in the air is twice the time to reach the highest point (since time ascending equals time descending):\[ T = 2 \times 0.30 = 0.60 \text{ seconds} \].

Key concepts

Initial Speed Calculation

When an object, like a flea, jumps straight up, calculating the initial speed is essential to understand how it propels itself upward. This initial speed is the velocity at which the flea leaves the ground before the effects of gravity slow it down to a stop at the highest point of the jump. To find this initial speed, we use a kinematic equation:- It relates the initial speed (\( u \)), final speed (\( v \)), acceleration (\( a \)), and distance traveled (\( s \)).- The equation is: \[ v^2 = u^2 + 2as \]- For a jump, the final velocity (\( v \)) at the peak height is 0 m/s because the flea stops rising and begins to fall.By substituting this and simplifying, we calculate the initial speed needed to reach a certain height. For example, if a flea jumps to 0.440 m, the initial speed can be calculated as follows:\[ u = \sqrt{2 \times 9.81 \times 0.440} \approx 2.94 \text{ m/s} \] This result shows that the flea had an initial speed of approximately 2.94 m/s when it left the ground.

Time of Flight

Time of flight refers to the duration an object stays in the air during its motion - from the moment it leaves the ground to when it returns. For vertical jumps, like a flea's, this time can be divided into two equal parts due to symmetry:- **Ascent**: Time taken to reach the peak point.- **Descent**: Time taken to fall back from the peak to the ground.We can compute the time to ascend using the equation:- \[ 0 = u - 9.81t \]- Rearranging gives: \[ t = \frac{u}{9.81} \] Where: - \( u \) is the initial speed - \( 9.81 \text{ m/s}^2 \) is the gravitational accelerationGiven an initial speed \( u \approx 2.94 \text{ m/s} \), the time to peak is:\[ t \approx \frac{2.94}{9.81} \approx 0.30 \text{ seconds} \]Therefore, the total time of flight is:\[ T = 2 \times 0.30 = 0.60 \text{ seconds} \]This calculation indicates the total air time for the flea round trip is around 0.60 seconds.

Acceleration Due to Gravity

The acceleration due to gravity is a constant force that pulls objects towards the Earth’s center. It is denoted by \( g \) and approximately equals 9.81 m/s². This value is crucial in kinematic equations when analyzing vertical motion.For a jumping object:- Gravity works against the initial upward motion, slowing it until it halts at the peak.- Once at the peak, gravity accelerates the object downwards back to the ground.Gravity's role in kinematics:- Increases the speed of the object in free fall.- Can be included in equations as: \[ v = u + at \] Where \( a = -9.81 \text{ m/s}^2 \) when moving upwards (since gravity opposes the motion), and \( a = 9.81 \text{ m/s}^2 \) when moving downwards.Understanding gravity helps predict how objects move through vertical space, like how a flea can successfully leap a certain height, remain in the air momentarily, and land.