Question

Two cars start 200 m apart and drive toward each other at a steady 10 m/s. On the front of one of them, an energetic grasshopper jumps back and forth between the cars (he has strong legs!) with a constant horizontal velocity of 15 m/s relative to the ground. The insect jumps the instant he lands, so he spends no time resting on either car. What total distance does the grasshopper travel before the cars hit?

Short answer

The grasshopper travels a total distance of 150 m before the cars collide.

Step-by-step solution

Determine the time until the cars collide

Since both cars are traveling towards each other, we can calculate the time it takes for them to collide by using the formula for relative speed. The relative speed of the two cars is the sum of their speeds: \(10 \text{ m/s} + 10 \text{ m/s} = 20 \text{ m/s}\). The distance between them is initially 200 m. So, we calculate the time it takes for them to collide by using: \(\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{200 \text{ m}}{20 \text{ m/s}} = 10 \text{ seconds}\).

Calculate the total distance the grasshopper travels

The grasshopper moves at a constant speed of 15 m/s. During the 10 seconds that the cars take to collide, the grasshopper is jumping continuously between the two cars. The total distance the grasshopper travels can be calculated as \(\text{Distance} = \text{Speed} \times \text{Time} = 15 \text{ m/s} \times 10 \text{ s} = 150 \text{ m}\).

Key concepts

Constant Velocity

Constant velocity is a key concept in understanding motion, especially in problems involving motion in one dimension. When an object moves with a constant velocity, it means the object covers equal distances in equal intervals of time without changing speed or direction.
For instance, in the exercise, the grasshopper maintains a steady speed of 15 meters per second. This is an example of constant velocity since it does not speed up or slow down, and always moves in a straight path, jumping back and forth.

• Definition: Velocity that does not change with time.
• Characteristics: No acceleration; both speed and direction remain unchanged.
• Formula: Distance traveled = Velocity x Time

Understanding the nature of constant velocity helps simplify calculations as it involves straightforward multiplication of speed and time.

Relative Speed

Relative speed is an important concept when analyzing how two or more objects move with respect to each other. It describes how fast one object is moving compared to another object.
This is crucial in the given exercise, where two cars start moving toward each other. The relative speed can be calculated by adding their individual speeds because they are moving in opposite directions towards each other.
For the cars:

• Car 1 Speed: 10 m/s
• Car 2 Speed: 10 m/s
• Relative Speed: 10 m/s + 10 m/s = 20 m/s

This tells us how fast the gap between them is closing. Relative speed simplifies complex scenarios, allowing us to treat two moving objects as one system, making calculations about their interactions easier to handle.

Distance and Displacement

Distance and displacement are fundamental concepts in physics often confused but distinctly different. In the given problem, they play a crucial role in analyzing the motion of the grasshopper.
Distance is the total path length traveled by the object, no matter the direction. For the grasshopper constantly jumping, the distance is how far it travels back and forth. Calculating it involves straightforward multiplication (speed x time).
Displacement, on the other hand, is concerned with the straight-line distance between the starting and ending points, plus direction. In this case, the grasshopper's displacement, when the cars collide, will be zero since it returns back and forth to the same points.

• Distance: Total length of the path traveled (e.g., 150 m for the grasshopper)
• Displacement: Change in position from start to finish (e.g., 0 m as it starts and ends at the cars)

Clearly distinguishing these concepts is vital in motion-related problems, enabling accurate calculations and understanding of an object's movement.