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Chapter 4: Problem 56

Is it possible for a helicopter to have an acceleration due east and a velocity due west? If so, what would be going on? If not, why not?

Short Answer

Expert verified
Yes, this is possible; the helicopter is slowing down while moving west.

Step by step solution

01

Understanding Concepts

To determine if it's possible for a helicopter to have an acceleration due east but a velocity due west, we first need to understand what velocity and acceleration mean. Velocity is the speed of an object in a particular direction. Acceleration is the rate of change of velocity over time, including changes in speed or direction.
02

Analyzing Directions

In this scenario, the helicopter has a velocity due west and an acceleration due east. Velocity due west means the helicopter is currently moving towards the west. Acceleration due east suggests that the helicopter is experiencing a force that is pushing/pulling it towards the east.
03

Considering Physical Motion

Even though the helicopter's velocity is directed west, an eastward acceleration implies the helicopter is slowing down as it moves west. Eventually, if the eastward acceleration continues, it may stop and potentially start moving east if the force continues to act on it.
04

Concluding Possibility

It is indeed possible for a helicopter to have an acceleration in one direction and velocity in the opposite direction. This means the helicopter is in the process of slowing down and could eventually reverse its direction.

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Key Concepts

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

Velocity and Acceleration
Motion in physics is not just about moving in one direction. It involves changes in speed and the direction of that motion, known as velocity, as well as changes in velocity, which is called acceleration.
Velocity describes both how fast an object is moving and in which direction. For example, a helicopter flying west at 50 mph has a velocity of 50 mph due west.
  • Velocity includes both speed and direction.
  • It is a vector quantity, meaning it has both magnitude and direction.
Acceleration, on the other hand, is the rate at which velocity changes over time. It can involve speeding up, slowing down, or changing direction. If a helicopter鈥檚 velocity decreases as it flies west due to an eastward force, this is acceleration.
  • Acceleration is also a vector quantity.
  • It explains how velocity changes - whether by speed or direction.
Directional Motion
When discussing motion, it's important to understand how direction plays a crucial role. A helicopter moving westward may encounter forces that push it eastward, affecting its movement.
Directional motion considers the influence of external forces that may not align with the current direction of travel. This is clearly shown when a helicopter moves west but experiences an eastward acceleration.
  • Motion direction can oppose force direction.
  • Eastward force on westward motion slows the westward motion.
Eventually, the helicopter may change its travel direction if the eastward acceleration continues, due to the persistent force eventually overcoming the westward velocity. This interplay of force and direction demonstrates the dynamic nature of motion.
Newton's Laws of Motion
Newton's Laws of Motion provide the framework for understanding how forces affect motion and are essential for explaining the initial scenario of a helicopter moving west, but accelerating east.
Newton's First Law, often called the law of inertia, states that an object will remain at rest or in uniform motion unless acted upon by a net force. This explains why the helicopter maintains its westward velocity unless acted upon by the eastward force.
  • Law of inertia: Continuation in state of motion.
  • Mass resists changes in motion.
Newton's Second Law connects force, mass, and acceleration, stating that the force on an object is equal to its mass times its acceleration (F = ma). This is crucial for understanding how the eastward force changes the helicopter's velocity.
  • Force leads to acceleration in the direction of the force.
  • The greater the force, the greater the change in motion.
Finally, Newton鈥檚 Third Law, which states that for every action, there is an equal and opposite reaction, helps us understand the interaction between the helicopter and the surrounding air pushing it against the direction of its motion.

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Most popular questions from this chapter

Find the angle between the following two vectors: $$ \begin{array}{l} \hat{\mathbf{x}}+2 \hat{\mathbf{y}}+3 \hat{\mathbf{z}} \\ 4 \hat{\mathbf{x}}+5 \hat{\mathbf{y}}+6 \hat{\mathbf{z}} \end{array} $$ \mathrm{\\{} ~ h w h i n t ~ \\{ h w h i n t : a n g l e b e t w e e n \\} ( a n s w e r ~ c h e c k ~ a v a i l a b l e ~ a t ~ l i g h t a n d m a t t e r . c o m ) ~

A gun is aimed horizontally to the west. The gun is fired, and the bullet leaves the muzzle at \(t=0\). The bullet's position vector as a function of time is \(\mathbf{r}=b \hat{\mathbf{x}}+c t \hat{\mathbf{y}}+d t^{2} \hat{\mathbf{z}}\), where \(b, c\), and \(d\) are positive constants. (a) What units would \(b, c\), and \(d\) need to have for the equation to make sense? (b) Find the bullet's velocity and acceleration as functions of time. (c) Give physical interpretations of \(b, c, d, \hat{\mathbf{x}}, \hat{\mathbf{y}}\), and \(\hat{\mathbf{z}}\).

A batter hits a baseball at speed \(v\), at an angle \(\theta\) above horizontal. (a) Find an equation for the range (horizontal distance to where the ball falls), \(R\), in terms of the relevant variables. Neglect air friction and the height of the ball above the ground when it is hit. \hwans\\{hwans:baseballrange\\} (b) Interpret your equation in the cases of \(\theta=0\) and \(\theta=90^{\circ}\). (c) Find the angle that gives the maximum range. Wwans\\{hwans:baseballrange\\}

A helicopter of mass \(m\) is taking off vertically. The only forces acting on it are the earth's gravitational force and the force, \(F_{a i r}\), of the air pushing up on the propeller blades. (a) If the helicopter lifts off at \(t=0\), what is its vertical speed at time \(t\) ? (b) Check that the units of your answer to part a make sense. (c) Discuss how your answer to part a depends on all three variables, and show that it makes sense. That is, for each variable, discuss what would happen to the result if you changed it while keeping the other two variables constant. Would a bigger value give a smaller result, or a bigger result? Once you've figured out this mathematical relationship, show that it makes sense physically. (d) Plug numbers into your equation from part a, using \(m=2300 \mathrm{~kg}, F_{a i r}=27000 \mathrm{~N}\), and \(t=4.0 \mathrm{~s}\). (answer check available at lightandmatter.com)

Mountain climbers with masses \(m\) and \(M\) are roped together while crossing a horizontal glacier when a vertical crevasse opens up under the climber with mass \(M\). The climber with mass \(m\) drops down on the snow and tries to stop by digging into the snow with the pick of an ice ax. Alas, this story does not have a happy ending, because this doesn't provide enough friction to stop. Both \(m\) and \(M\) continue accelerating, with \(M\) dropping down into the crevasse and \(m\) being dragged across the snow, slowed only by the kinetic friction with coefficient \(\mu_{k}\) acting between the ax and the snow. There is no significant friction between the rope and the lip of the crevasse. (a) Find the acceleration \(a\).(answer check available at lightandmatter.com) (b) Check the units of your result. (c) Check the dependence of your equation on the variables. That means that for each variable, you should determine what its effect on \(a\) should be physically, and then what your answer from part a says its effect would be mathematically.
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