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Chapter 2: Problem 64

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

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

Step by step solution

01

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: \(\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{200 \text{ m}}{20 \text{ m/s}} = 10 \text{ seconds}\).
02

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 \(\text{Distance} = \text{Speed} \times \text{Time} = 15 \text{ m/s} \times 10 \text{ s} = 150 \text{ m}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A helicopter carrying Dr. Evil takes off with a constant upward acceleration of 5.0 m/s\(^2\). Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of the helicopter. Assume that the helicopter is in free fall after its engine is shut off, and ignore the effects of air resistance. (a) What is the maximum height above ground reached by the helicopter? (b) Powers deploys a jet pack strapped on his back 7.0 s after leaving the helicopter, and then he has a constant downward acceleration with magnitude 2.0 m/s\(^2\). How far is Powers above the ground when the helicopter crashes into the ground?

During your summer internship for an aerospace company, you are asked to design a small research rocket. The rocket is to be launched from rest from the earth's surface and is to reach a maximum height of 960 m above the earth's surface. The rocket's engines give the rocket an upward acceleration of 16.0 m/s\(^2\) during the time \(T\) that they fire. After the engines shut off, the rocket is in free fall. Ignore air resistance. What must be the value of \(T\) in order for the rocket to reach the required altitude?

The rocket-driven sled \(\textit{Sonic Wind No. 2,}\) used for investigating the physiological effects of large accelerations, runs on a straight, level track 1070 m (3500 ft) long. Starting from rest, it can reach a speed of 224 m/s(500 mi/h) in 0.900 s. (a) Compute the acceleration in m/s\(^2\), assuming that it is constant. (b) What is the ratio of this acceleration to that of a freely falling body (\(g\))? (c) What distance is covered in 0.900 s? (d) A magazine article states that at the end of a certain run, the speed of the sled decreased from 283 m/s (632 mi/h) to zero in 1.40 s and that during this time the magnitude of the acceleration was greater than 40\(g\). Are these figures consistent?

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 2.80 m/s\(^2\). At the same instant a truck, traveling with a constant speed of 20.0 m/s, overtakes and passes the car. (a) How far beyond its starting point does the car overtake the truck? (b) How fast is the car traveling when it overtakes the truck? (c) Sketch an \(x-t\) graph of the motion of both vehicles. Take \(x=\) 0 at the intersection. (d) Sketch a \(v_x-t\) graph of the motion of both vehicles.

Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. (a) If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? (b) If the slower stone reaches a maximum height of \(H\), how high (in terms of \(H\)) will the faster stone go? Assume free fall.
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