Question

By trial and error, a frog learns that it can leap a maximum horizontal distance of \(1.30 \mathrm{~m}\). If, in the course of an hour, the frog spends \(20.0 \%\) of the time resting and \(80.0 \%\) of the time performing identical jumps of that maximum length, in a straight line, what is the distance traveled by the frog?

Short answer

Answer: The expression for the total distance traveled by the frog is \(\frac{1.30 \times 2880}{t} \mathrm{~m}\).

Step-by-step solution

Calculate time spent jumping

In an hour, the frog spends 80.0% of the time jumping and 20.0% of the time resting. Since there are 3600 seconds in an hour, the frog spends 0.8 * 3600 seconds jumping. Calculate the time spent jumping: \((0.8) (3600 \,\mathrm{seconds}) = 2880 \,\mathrm{seconds}\)

Calculate number of jumps

Since the frog spends 2880 seconds jumping and performs identical jumps, we need to find out how long it takes for the frog to perform a single jump. Let's assume it takes 't' seconds for a single jump. Then in 2880 seconds, the frog can make 2880/t jumps.

Calculate distance of one jump

The frog can leap a maximum horizontal distance of 1.30 m in a single jump. So the distance covered in one jump is 1.30 m.

Calculate total distance traveled

To find the total distance traveled by the frog, we can multiply the distance of one jump and the number of jumps. As we don't know the exact number of jumps (2880/t), let's represent it as: \(\text{Total distance} = 1.30 \times \frac{2880}{t} \) However, we are not given the time taken for a single jump (t). Since we can't calculate the exact distance, we can only express it in terms of 't': \(\text{Total distance} = \frac{1.30 \times 2880}{t} \mathrm{~m}\)

Key concepts

Kinematics

Understanding kinematics is essential when dealing with motion problems in physics. Kinematics refers to the branch of mechanics that focuses on the description of motion without considering the causes. It involves the concepts of displacement, velocity, acceleration, and time.

When observing our frog's leaps, these concepts come into play. The maximum horizontal distance of a leap (1.30 meters) can be considered as displacement. If we knew the time 't' per jump, we could calculate the leap's velocity. However, since kinematics also deals with uniformly accelerated motion, it's important to know that the frog's leap probably involves acceleration due to gravity and potentially initial velocity at take-off.

This understanding helps us analyze the frog's movement and later calculate the total distance traveled. The clearer the concept of kinematics is, the easier it will be to tackle these types of problems.

Projectile Motion

Projectile motion is a form of motion experienced by an object that is launched near the Earth's surface and moves along a curved path under the action of gravity only. Our frog's leap is a perfect real-world example of projectile motion. This type of motion is characterized by a horizontal displacement and a vertical displacement simultaneously.

Due to gravity, the frog's leap has a parabolic trajectory, reaching a maximum height before descending back to the ground. To simplify calculations, air resistance is often neglected. In an ideal projectile motion scenario, there are two key components: horizontal velocity (constant) and vertical velocity (changes due to gravity).

Understanding these components would allow us to calculate the time 't' for one jump. With that information, we could determine how many jumps the frog makes and the total distance traveled—showing the direct link between kinematics and projectile motion.

Uniformly Accelerated Motion

Uniformly accelerated motion describes when an object's velocity changes by an equal amount in each equal time period. Gravity is the perfect example as it accelerates all objects at approximately 9.81 m/s² towards the Earth's center. In our problem, the frog is subjected to uniformly accelerated motion during its leaps.

The vertical component of the frog's jumps will be under uniformly accelerated motion due to gravity. Interestingly, even if we consider the horizontal motion to be constant during the jump (barring air resistance), the vertical component will still be accelerating until the peak of the arc, and then decelerating as the frog lands. The concept of uniformly accelerated motion is crucial in calculating the time 't' for each leap if we knew the initial take-off speed and the height of the leap.

Combining knowledge of kinematics, projectile motion, and uniformly accelerated motion allows us to tackle complex motion problems by breaking them down into simpler, more manageable parts.