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Chapter 3: Problem 20

Two cannonballs are shot in sequence from a cannon, into the air, with the same muzzle velocity, at the same launch angle. Based on their trajectory and range, how can you tell which one is made of lead and which one is made of wood. If the same cannonballs where launched in vacuum, what would the answer be?

Short Answer

Expert verified
Answer: The lead cannonball would have a longer range and a trajectory closer to the ideal parabolic motion due to its greater mass and density, which results in a smaller deceleration relative to its mass due to air resistance compared to the lighter wooden cannonball.

Step by step solution

01

Analyze the forces acting on the cannonballs

When the cannonballs are launched, the primary forces acting on them are gravity and air resistance. Since they have the same launch angle and velocity, we can differentiate between the two based on how much air resistance affects their motion. As lead has greater mass and density, it will experience a smaller deceleration relative to its mass due to air resistance compared to the lighter wooden cannonball.
02

Determine the effect of air resistance on the trajectory

The air resistance will have a greater impact on the trajectory of the lighter wooden cannonball than on the heavier lead one, causing it to have a shorter range and a trajectory that differs from the ideal parabolic path more than the lead cannonball would. Thus, the wooden cannonball's trajectory will deviate more from the ideal parabolic motion compared to the lead cannonball.
03

Compare the trajectory and range

When comparing the trajectory and range of the two cannonballs, the one made of lead will have a longer range and a trajectory closer to the ideal parabolic motion. The wooden cannonball will have a shorter range and a trajectory that deviates more from the ideal parabolic path. This is due to the larger effect of air resistance on the lighter wooden cannonball.
04

Determine the result in vacuum

If the same cannonballs were launched in a vacuum, there would be no air resistance to act on them. Therefore, both cannonballs would follow an ideal parabolic trajectory, and their range would only be determined by their initial launch angle and velocity. Since both launch angle and velocity are identical for the two cannonballs, they would have the same trajectory and range in a vacuum. So you would not be able to tell which one is made of lead and which one is made of wood based on their trajectory and range in a vacuum.

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Most popular questions from this chapter

You passed the salt and pepper shakers to your friend at the other end of a table of height \(0.85 \mathrm{~m}\) by sliding them across the table. They both missed your friend and slid off the table, with velocities of \(5 \mathrm{~m} / \mathrm{s}\) and \(2.5 \mathrm{~m} / \mathrm{s}\), respectively. a) Compare the times it takes the shakers to hit the floor. b) Compare the distance that each shaker travels from the edge of the table to the point it hits the floor.

A projectile is launched from the top of a building with an initial velocity of \(30 \mathrm{~m} / \mathrm{s}\) at an angle of \(60^{\circ}\) above the horizontal. The magnitude of its velocity at \(t=5 \mathrm{~s}\) after the launch is a) \(-23.0 \mathrm{~m} / \mathrm{s}\) c) \(15.0 \mathrm{~m} / \mathrm{s}\) e) \(50.4 \mathrm{~m} / \mathrm{s}\) b) \(7.3 \mathrm{~m} / \mathrm{s}\) d) \(27.5 \mathrm{~m} / \mathrm{s}\)

An arrow is shot horizontally with a speed of \(20 \mathrm{~m} / \mathrm{s}\) from the top of a tower \(60 \mathrm{~m}\) high. The time to reach the ground will be a) \(8.9 \mathrm{~s}\) c) \(3.5 \mathrm{~s}\) e) \(1.0 \mathrm{~s}\) b) \(7.1 \mathrm{~s}\) d) \(2.6 \mathrm{~s}\)

10.0 seconds after being fired, a cannonball strikes a point \(500 . \mathrm{m}\) horizontally from and \(100 . \mathrm{m}\) vertically above the point of launch. a) With what initial velocity was the cannonball launched? b) What maximum height was attained by the ball? c) What is the magnitude and direction of the ball's velocity just before it strikes the given point?

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?
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