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Chapter 6: Problem 26

Under the influence of its drive force, a snowmobile is moving at a constant velocity along a horizontal patch of snow. When the drive force is shut off, the snowmobile coasts to a halt. The snowmobile and its rider have a mass of 136 kg. Under the influence of a drive force of 205 N, it is moving at a constant velocity whose magnitude is 5.50 m/s. The drive force is then shut off. Find (a) the distance in which the snowmobile coasts to a halt and (b) the time required to do so.

Short Answer

Expert verified
The snowmobile coasts to a halt over a distance of 10.01 meters in approximately 3.64 seconds.

Step by step solution

01

Identify Forces Acting on Snowmobile

The snowmobile moves at a constant velocity under a drive force of 205 N. This implies that the force of friction is equal in magnitude but opposite in direction to the drive force when moving at constant velocity. Hence, the frictional force, when the drive stops, is 205 N.
02

Calculate the Deceleration Due to Friction

Since the drive force stops, only the friction force affects the motion. Using Newton's second law, the deceleration \( a \) is found by \( a = \frac{f}{m} = \frac{205 \text{ N}}{136 \text{ kg}} \approx 1.51 \text{ m/s}^2 \).
03

Use Kinematic Equation to Find Distance

Using the equation \( v^2 = u^2 + 2as \), with final velocity \( v = 0 \), initial velocity \( u = 5.50 \text{ m/s} \), and \( a = -1.51 \text{ m/s}^2 \), solve for distance \( s \). \[ 0 = (5.50)^2 + 2(-1.51)s \] Simplifying, \( s = \frac{(5.50)^2}{2 \times 1.51} \approx 10.01 \text{ meters} \).
04

Calculate Time Using Another Kinematic Equation

Use the kinematic equation \( v = u + at \) to find time \( t \). Again, with \( v = 0 \), \( u = 5.50 \text{ m/s} \), and \( a = -1.51 \text{ m/s}^2 \), solve for \( t \): \[ 0 = 5.50 + (-1.51)t \] Simplification yields \( t = \frac{5.50}{1.51} \approx 3.64 \text{ seconds} \).

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Key Concepts

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

kinematic equations
Kinematic equations play a crucial role in solving problems related to motion under uniform acceleration. They are often used in physics to predict the future position or velocity of an object moving with constant acceleration. These equations make it possible to calculate various elements of motion without having to directly measure them.
For the snowmobile, we applied two kinematic equations:
  • To find the distance the snowmobile traveled until it stopped, we used the equation \( v^2 = u^2 + 2as \). Here, \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the deceleration, and \( s \) is the distance. The final velocity is zero because the snowmobile stops.

  • To determine how long it took for the snowmobile to stop, we used the equation \( v = u + at \), which relates the final and initial velocities, acceleration, and time.
Understanding these equations allows one to predict how long it takes for an object to stop under a specific force, and how far it will travel in this timeframe.
frictional force
Frictional force significantly influences motion, especially when a driving force is removed. It originates from the interaction between surfaces in contact and tends to oppose motion, acting opposite to the direction of movement.
In the snowmobile example, when the drive force is cut off, the frictional force becomes the only force acting against its motion. Initially, the frictional force exactly balances the drive force (205 N) when moving at constant velocity. Therefore, when the drive force is halted, the frictional force remains the same and acts alone to stop the snowmobile.
By applying Newton's second law \( F = ma \), where \( F \) is the frictional force, \( m \) is the mass, and \( a \) is acceleration (or deceleration in this case), we can determine the deceleration caused by this frictional force. Understanding the role of friction helps explain why objects do not continue moving indefinitely when no longer propelled by a force.
deceleration
Deceleration is the rate at which an object slows down, which is essentially negative acceleration. It's vital to distinguish between acceleration and deceleration, as they affect motion in opposite ways.
In the context of the snowmobile, deceleration occurs once the drive force is removed, and the opposing frictional force takes over. The deceleration can be calculated by dividing the frictional force by the mass of the snowmobile, resulting in \( a = \frac{205 \text{ N}}{136 \text{ kg}} \approx 1.51 \text{ m/s}^2 \).
Knowing the deceleration helps in predicting how quickly an object will stop. By applying kinematic equations alongside this deceleration value, we can find out not only the distance the snowmobile will coast until it halts but also the time it will take.
Deceleration is a key factor in understanding how forces interact with objects to change their state of motion.

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