Question

By trial and error, a frog learns that it can leap a maximum horizontal distance of \(1.3 \mathrm{~m}\). If, in the course of an hour, the frog spends \(20 \%\) of the time resting and \(80 \%\) of the time performing identical jumps of that maximum length, in a straight line, what is the distance traveled by the frog?

Short answer

Question: In one hour, a frog travels continuously by leaping. It spends 80% of the time leaping, with each leap having a maximum horizontal distance of 1.3 meters. Calculate the total distance traveled by the frog in one hour. Answer: The frog travels a total distance of 3744 meters.

Step-by-step solution

Calculate the amount of time spent leaping

We know that the frog spends 80% of the time leaping. Since the total time is one hour, we can find the time spent leaping by multiplying the total time by the given percentage. As a decimal, 80% is 0.8. So, the amount of time spent leaping is: \(1 \text{ hour} \times 0.8 = 0.8 \text{ hours}\)

Convert hours to seconds

To determine the number of leaps, we need to convert the time spent leaping in hours to seconds. There are 3600 seconds in an hour, so we can convert the time as follows: \(0.8 \text{ hours} \times 3600 \frac{\text{seconds}}{\text{hour}} = 2880 \text{ seconds}\)

Calculate the number of leaps

The frog performs identical jumps continuously during the time spent leaping. We are not given the duration of each leap, but we are asked to find the distance traveled in one hour. So, let's assume that the frog completes one leap per second. Thus, the number of leaps performed in the 2880 seconds is: \(2880 \text{ leaps}\)

Calculate the total distance traveled

Now that we know that the frog performs 2880 leaps, each with a maximum horizontal distance of 1.3 meters, we can calculate the total distance the frog travels by multiplying these values together: \(2880 \text{ leaps} \times 1.3 \frac{\text{meters}}{\text{leap}} = 3744 \text{ meters}\) Therefore, the frog travels a total distance of 3744 meters.

Key concepts

Kinematics

Kinematics is a branch of physics that studies the motion of objects without considering the forces that cause them. Understanding kinematics is crucial when analyzing how entities like frogs move in their environment. In this particular problem, the frog’s jumping motion can be described using kinematic principles.

The frog's leap represents a projectile motion, where its horizontal component is of interest. Since only horizontal displacement is considered in this problem, it simplifies calculations by disregarding vertical movement and air resistance. Every jump the frog makes covers a distance of 1.3 meters. This consistent motion forms the basis of solving how far the frog travels over a period.

When dealing with such scenarios, it's key to separate different dimensions of movement (vertical and horizontal) and to focus on the parameters you need to solve the problem. This clarity streamlines understanding exactly how kinematics is applied to real-world examples like the frog’s leap.

Distance Calculation

Calculating distances in projectile motion requires an understanding of the individual leaps an object makes. In this scenario, the frog's maximum jump length is known to be 1.3 meters.

To find the total distance traveled, you need to determine the total number of leaps. Once this is known, you multiply the number of leaps by the distance covered in a single leap:

• Number of leaps: 2880
• Distance per leap: 1.3 meters

Thus, the calculation for total distance is straightforward: the number of leaps times the distance of each leap. This gives us the total distance of 3744 meters.

Understanding how simple multiplication can provide the total displacement in repeated motion is vital for handling problems involving consistent repetitive movements.

Time Conversion

Time conversion is an essential skill, especially when dealing with scenarios involving time proportions and motion. In this problem, the frog spends 80% of its time jumping during an hour. Thus, converting that time from hours to seconds helps in determining the number of jumps made.

Start by converting the percentage to a decimal, where 80% becomes 0.8. Multiply this by the total time of 1 hour. Since the calculation needs to determine seconds, convert this time amount into seconds:

• 1 hour = 3600 seconds
• Leap time: 0.8 hours = 2880 seconds

It is important to remember the conversion factor between hours and seconds because many real-world problems require converting different time units to compute subsequent steps in a calculation.

Time conversion ensures that you have the correct time units to perform further calculations, especially in physics where time often needs to be in smaller units like seconds.