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Chapter 3: Problem 13

In ideal projectile motion, when the positive \(y\) -axis is chosen to be vertically upward, the \(y\) -component of the velocity of the object during the ascending part of the motion and the \(y\) -component of the velocity during the descending part of the motion are, respectively, a) positive, negative. c) positive, positive. b) negative, positive. d) negative, negative.

Short Answer

Expert verified
Answer: The y-components of the velocity during the ascending and descending parts of an object in ideal projectile motion are positive and negative, respectively. This is because the object is moving upward against the force of gravity during the ascending part, and downward due to the force of gravity during the descending part.

Step by step solution

01

Analyze the ascending part of the motion

During the ascending part, the object is moving upward against the force of gravity. As a result, the y-component of the velocity is decreasing, but the object is still moving upward, so the velocity is positive.
02

Analyze the descending part of the motion

During the descending part, the object is moving downward due to the force of gravity. Hence, the y-component of the velocity is increasing in the downward direction, meaning it is negative.
03

Identify the correct option

Based on the analysis above, the y-component of the velocity during ascending is positive, and during descending, it is negative. Therefore, the correct option is: a) positive, negative.

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Key Concepts

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

Velocity Components
In projectile motion, velocity can be understood through its components: horizontal and vertical. The projectile鈥檚 velocity is not just a single speed but made up of these two parts working together.
  • **Horizontal Component**: This remains constant throughout the motion if we ignore air resistance; it doesn't change because no force acts horizontally in ideal conditions.
  • **Vertical Component**: This changes during the motion due to the force of gravity. As the object ascends, this component decreases, and as it descends, it increases negatively.
The initial velocity vector can be split into these components using trigonometry, especially by knowing the angle of projection. These components help predict the projectile鈥檚 path and are crucial in understanding how the projectile travels through the air.
Ascending and Descending Motion
The motion of a projectile can be divided into two primary phases: ascending and descending, which are critically governed by gravity. During **ascending motion**:
  • The projectile moves against gravity. Its vertical component of velocity decreases until it reaches the peak.
  • At the peak, this vertical velocity becomes zero, marking the transition from ascending to descending.
In the **descending motion**:
  • The projectile moves in the same direction as gravity. The vertical component of velocity becomes negative and begins to increase in magnitude.
  • This phase continues until the projectile impacts the ground.
This cyclical nature highlights how gravity uniquely affects each phase of projectile motion.
Force of Gravity
Gravity is a crucial force in projectile motion. It consistently acts downward, influencing both ascending and descending parts of the motion.
  • **On Ascending**: Gravity slows down the upward motion, reducing the vertical velocity until motion stops at the peak.
  • **On Descending**: Gravity accelerates the object downward, increasing the negative vertical velocity component.
The gravitational force remains constant throughout the motion, which makes the change in velocity predictable. Its role in creating a symmetrical path for many projectiles is why understanding gravity is vital in calculating accurate motion paths. This predictability only breaks in real-world scenarios where air resistance and other forces come into play. However, in an ideal scenario often studied in physics, gravity is the sole vertical force affecting motion.

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Most popular questions from this chapter

In a projectile motion, the horizontal range and the maximum height attained by the projectile are equal. a) What is the launch angle? b) If everything else stays the same, how should the launch angle, \(\theta_{0},\) of a projectile be changed for the range of the projectile to be halved?

The air speed indicator of a plane that took off from Detroit reads \(350 . \mathrm{km} / \mathrm{h}\) and the compass indicates that it is heading due east to Boston. A steady wind is blowing due north at \(40.0 \mathrm{~km} / \mathrm{h}\). Calculate the velocity of the plane with reference to the ground. If the pilot wishes to fly directly to Boston (due east) what must the compass read?

In ideal projectile motion, when the positive \(y\) -axis is chosen to be vertically upward, the \(y\) -component of the acceleration of the object during the ascending part of the motion and the \(y\) -component of the acceleration during the descending part of the motion are, respectively, a) positive, negative. c) positive, positive. b) negative, positive. d) negative, negative.

In a three-dimensional motion, the \(x-, y-\), and \(z\) coordinates of the object as a function of time are given by \(x(t)=\frac{\sqrt{2}}{2} t, \quad y(t)=\frac{\sqrt{2}}{2} t,\) and \(z(t)=-4.9 t^{2}+\sqrt{3} t\) Describe the motion and the trajectory of the object in an \(x y z\) coordinate system.

Neglect air resistance for the following. A soccer ball is kicked from the ground into the air. When the ball is at a height of \(12.5 \mathrm{~m},\) its velocity is \((5.6 \hat{x}+4.1 \hat{y}) \mathrm{m} / \mathrm{s}\). a) To what maximum height will the ball rise? b) What horizontal distance will be traveled by the ball? c) With what velocity (magnitude and direction) will it hit the ground?
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