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Chapter 3: Problem 29

A key falls from a bridge that is \(45 \mathrm{~m}\) above the water. It falls directly into a model boat, moving with constant velocity, that is \(12 \mathrm{~m}\) from the point of impact when the key is released. What is the speed of the boat?

Short Answer

Expert verified
The speed of the boat is approximately 3.96 m/s.

Step by step solution

01

Determine the time it takes for the key to fall

Use the equation for the distance traveled under constant acceleration due to gravity: \[ d = \frac{1}{2}gt^2 \] Here, \( d = 45 \text{ m} \) and \( g = 9.8 \text{ m/s}^2 \). Solving for \( t \): \[ 45 = \frac{1}{2} \times 9.8 \times t^2 \] \[ 45 = 4.9 \times t^2 \] \[ t^2 = \frac{45}{4.9} \] \[ t^2 = 9.18 \] \[ t = \text{sqrt}(9.18) \] \[ t \rightarrow 3.03 \text{ s} \]
02

Determine how far the boat travels in the time t

The key takes 3.03 seconds to fall. To find the speed of the boat, use the formula for constant velocity: \[ v = \frac{d}{t} \] Here, \( d \rightarrow 12 \text{ m} \) and \( t \rightarrow 3.03 \text{ s} \). Solving for \( v \): \[ v = \frac{12}{3.03} \] \[ v \rightarrow 3.96 \text{ m/s} \]

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Key Concepts

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

constant acceleration
In kinematics, constant acceleration refers to a scenario where the rate of change of velocity remains the same over time. This is often seen in objects falling under the influence of gravity, which has a constant acceleration of approximately \( 9.8 \text{ m/s}^2 \).

The key formula related to constant acceleration in this problem is \[ d = \frac{1}{2}gt^2 \].
Here:
  • \( d \) is the distance the object falls,
    \( g \) represents the acceleration due to gravity, and
    \( t \) is the time it takes to fall.
The acceleration due to gravity is a fundamental concept in physics and is crucial in solving motion problems.
velocity calculation
Velocity is a vector quantity that refers to the rate at which an object changes its position. It has both magnitude (speed) and direction. In the context of this problem, the boat's velocity is constant, allowing us to use a straightforward formula to find it.

The key formula is \[ v = \frac{d}{t} \]
In this formula,
  • \( v \) represents the velocity,
    \( d \) is the distance traveled, and
    \( t \) is the time taken.

We used this formula to determine the speed of the boat, given that the key takes 3.03 seconds to fall and the boat was initially 12 meters away from the point of impact. By substituting these values, we calculated the boat's velocity to be approximately \( 3.96 \text{ m/s} \).

Remember, because velocity is a vector, it's important to consider the direction as well. However, in this scenario, we are concerned only with the magnitude, as the direction remains constant.
motion equations
Motion equations are used to describe the relationship between displacement, velocity, acceleration, and time. They are fundamental in kinematics. This exercise involves constant acceleration, allowing us to use specific motion equations effectively.

In solving the problem, we primarily used the following kinematic equations:
  • For the key's fall: \[ d = \frac{1}{2}gt^2 \]
  • For the boat's speed: \[ v = \frac{d}{t} \]

These equations allow us to break down the problem step by step:
  • First, we calculated the time it takes for the key to fall 45 meters using \[ d = \frac{1}{2}gt^2 \].
    By solving for \( t \), we determined the fall time to be approximately 3.03 seconds.
  • Next, we used \[ t \] to find out the distance the boat travels in that time and subsequently calculated its constant velocity using \[ v = \frac{d}{t} \].

Practicing these equations helps in understanding how motion works and in solving more complex problems in physics.

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Most popular questions from this chapter

Your roommate peeks over your shoulder while you are reading a physics text and notices the following sentence: "In free fall the acceleration is always \(g\) and always straight downward regardless of the motion." Your roommate finds this peculiar and raises three objections: (a) If I drop a balloon or a feather, it doesn't fall nearly as fast as a brick. (b) Not everything falls straight down; if I throw a ball it can go sideways. (c) If I hold a wooden ball in one hand and a steel ball in the other, I can tell that the steel ball is being pulled down much more strongly than the wooden one. It will probably fall faster. How would you respond to these statements? Discuss the extent to which they invalidate the quoted statement. If they don't invalidate the statement, explain why.

Water drips from the nozzle of a shower onto the floor \(200 \mathrm{~cm}\) below. The drops fall at regular (equal) intervals of time, the first drop striking the floor at the instant the fourth drop begins to fall. Find the locations of the second and third drops when the first strikes the floor.

An \(85 \mathrm{~kg}\) man lowers himself to the ground from a height of \(10.0 \mathrm{~m}\) by holding onto a rope that runs over a frictionless pulley to a \(65 \mathrm{~kg}\) sandbag. With what speed does the man hit the ground if he started from rest?

A Frenchman, filling out a form, writes \(" 78 \mathrm{~kg}\) " in the space marked poids (weight). However weight is a force and \(\mathrm{kg}\) is a mass unit. What do the French (among others) have in mind when they use mass to report their weight? Why don't they report their weight in newtons? How many newtons does this Frenchman weigh? How many pounds?

Two An \(80 \mathrm{~kg}\) person is parachuting and experiencing a downward acceleration of \(2.5 \mathrm{~m} / \mathrm{s}^{2}\). The mass of the parachute is \(5.0 \mathrm{~kg}\). (a) What is the upward force on the open parachute from the air? (b) What is the downward force on the parachute from the person?
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