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Chapter 4: Problem 93

A penguin slides at a constant velocity of \(1.4 \mathrm{m} / \mathrm{s}\) down an icy incline. The incline slopes above the horizontal at an angle of \(6.9^{\circ} .\) At the bottom of the incline, the penguin slides onto a horizontal patch of ice. The coefficient of kinetic friction between the penguin and the ice is the same for the incline as for the horizontal patch. How much time is required for the penguin to slide to a halt after entering the horizontal patch of ice?

Short Answer

Expert verified
The penguin takes approximately 1.68 seconds to stop.

Step by step solution

01

Identify Forces Acting on Penguin

There is a kinetic friction force acting against the penguin on the horizontal ice patch, which is given by the formula \( f_k = \mu_k \cdot m \cdot g \), where \( \mu_k \) is the coefficient of kinetic friction, \( m \) is the mass of the penguin, and \( g \) is the acceleration due to gravity. However, since the velocity was constant on the incline, the friction is equal to the component of gravity along the incline.
02

Calculate Acceleration Due to Friction

Since the friction force has the same magnitude on both surfaces, it equals \( \mu_k \cdot m \cdot g \). The corresponding acceleration \( a \) decelerating the penguin can be found using Newton's second law: \( a = \mu_k \cdot g \), but because \( \mu_k \cdot g = g \cdot \sin(\theta) \) on the slope, \( a = g \cdot \sin(\theta) \).
03

Determine Time to Stop

The time \( t \) required to stop the penguin can be calculated using the kinematic equation \( v = u + at \), where \( v = 0 \) (final velocity), \( u = 1.4 \mathrm{m/s} \) (initial velocity), and \( a = - g \cdot \sin(\theta) \) (since it decelerates). Thus, \( 0 = 1.4 \mathrm{m/s} - (9.81 \mathrm{m/s^2} \cdot \sin(6.9^{\circ})) \cdot t \). Solve for \( t \).
04

Solve for \( t \)

Rearrange the equation \( 0 = 1.4 \mathrm{m/s} - (9.81 \mathrm{m/s^2} \cdot \sin(6.9^{\circ})) \cdot t \) to find \( t = \frac{1.4}{9.81 \cdot \sin(6.9^{\circ})} \). Calculate to find the value of \( t \).

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Key Concepts

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

Constant Velocity
When an object moves at a constant velocity, this means that its speed and direction remain unchanged. In the case of the penguin sliding down the icy incline, it exhibits constant velocity of 1.4 m/s. This indicates no net force acts on the penguin down the slope.
  • Think of driving a car at 60 mph on a straight road; you keep the speed constant, with no acceleration or deceleration.
  • For the penguin, gravity pulls it down the incline, while kinetic friction acts against it, balancing the forces so there is no change in velocity.
This concept helps us understand how the penguin maintained its speed while on the slope, making it easier to determine the changes once the incline ends.
Incline Angle
The incline angle, given here as 6.9 degrees, is crucial in determining the forces on the sloping surface. This angle helps calculate the force components acting along the incline, particularly the component of gravitational force.
  • The gravitational force splits into two components on an incline: parallel and perpendicular to the surface.
  • The parallel component, responsible for the slide, can be calculated using: \( F_{ ext{parallel}} = m imes g imes \sin(\theta) \).
  • Knowing this angle allows us to find how gravity affects the penguin's slide.
Understanding these force components is vital for calculating the deceleration when the penguin transitions to flat terrain.
Deceleration
Deceleration describes the decrease in velocity, crucial once the penguin hits the horizontal patch of ice. Initially moving due to momentum, the penguin gradually slows because of kinetic friction.
  • The deceleration is caused by the frictional force opposing the motion.
  • This force is given by \( f_k = \mu_k \times m \times g \).
  • Without the downhill gravitational pull, the friction from ice reduces its velocity to zero, halting the penguin.
Effectively, deceleration is a negative acceleration, indicating a decrease in speed over time, which we determine using the kinematic equations.
Kinematic Equations
Kinematic equations describe an object's motion under constant acceleration. For the penguin, these equations are necessary to find the stopping time on the horizontal ice patch.
  • The specific equation used is \( v = u + at \).
  • Here, initial velocity \( u = 1.4 \, \text{m/s} \), final velocity \( v = 0 \, \text{m/s} \), and acceleration \( a = -g \sin(\theta) \).
  • Plugging in these values into \( 0 = 1.4 - (9.81 \times \sin(6.9^{\circ})) \times t \) allows solving for \( t \), the time to stop.
Understanding this application of the kinematic equations enables us to see how the change from angle to flat friction impacts the penguin's motion.

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Most popular questions from this chapter

An 81 -kg baseball player slides into second base. The coefficient of kinetic friction between the player and the ground is \(0.49 .\) (a) What is the magnitude of the frictional force? (b) If the player comes to rest after \(1.6 \mathrm{s}\), what was his initial velocity?

A rocket of mass \(4.50 \times 10^{5} \mathrm{kg}\) is in flight. Its thrust is directed at an angle of \(55.0^{\circ}\) above the horizontal and has a magnitude of \(7.50 \times 10^{6} \mathrm{N} .\) Find the magnitude and direction of the rocket's acceleration. Give the direction as an angle above the horizontal.

Scientists are experimenting with a kind of gun that may eventually be used to fire payloads directly into orbit. In one test, this gun accelerates a \(5.0-\mathrm{kg}\) projectile from rest to a speed of \(4.0 \times 10^{3} \mathrm{m} / \mathrm{s} .\) The net force accelerating the projectile is \(4.9 \times 10^{5} \mathrm{N}\). How much time is required for the projectile to come up to speed?

A billiard ball strikes and rebounds from the cushion of a pool table perpendicularly. The mass of the ball is 0.38 kg. The ball approaches the cushion with a velocity of \(+2.1 \mathrm{m} / \mathrm{s}\) and rebounds with a velocity of \(-2.0 \mathrm{m} / \mathrm{s} .\) The ball remains in contact with the cushion for a time of \(3.3 \times 10^{-3} \mathrm{s} .\) What is the average net force (magnitude and direction) exerted on the ball by the cushion?

A skater with an initial speed of \(7.60 \mathrm{m} / \mathrm{s}\) stops propelling himself and begins to coast across the ice, eventually coming to rest. Air resistance is negligible. (a) The coefficient of kinetic friction between the ice and the skate blades is \(0.100 .\) Find the deceleration caused by kinetic friction. (b) How far will the skater travel before coming to rest?
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