Chapter 4: Problem 11
A hockey puck with mass 0.160 kg is at rest at the origin (\(x =\) 0) on the horizontal, frictionless surface of the rink. At time \(t =\) 0 a player applies a force of 0.250 N to the puck, parallel to the \(x\)-axis; she continues to apply this force until \(t =\) 2.00 s. (a) What are the position and speed of the puck at \(t =\) 2.00 s? (b) If the same force is again applied at \(t =\) 5.00 s, what are the position and speed of the puck at \(t =\) 7.00 s?
Short Answer
Step by step solution
Understand the Problem
Calculate Acceleration
Calculate Velocity at t = 2.00 s
Calculate Position at t = 2.00 s
Resume Force Application at t = 5.00 s
Calculate Position and Velocity at t = 5.00 s
Calculate New Velocity at t = 7.00 s
Calculate New Position at t = 7.00 s
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
constant acceleration equations
To find acceleration, we use Newton's Second Law of Motion, which states that the acceleration is the force applied divided by the mass of the object: \( a = \frac{F}{m} \).
Once we know the acceleration, we can calculate the velocity at any given time with the equation \( v = u + at \), where \( u \) is the initial velocity.
- This equation tells us that velocity depends linearly on time and directly on acceleration.
- The initial velocity \( u \) is assumed to be zero if the object starts from rest.
These equations allow us to solve many problems related to moving objects efficiently under the influence of consistent forces.
kinematics
We start by identifying initial conditions such as the initial position and velocity. In our exercise, the puck starts at rest at the origin. This simplifies our calculations as the initial conditions are zero for this specific context.
Once a force is applied, the puck starts accelerating, and kinematics allows us to calculate how far it travels and at what speed it moves using the constant acceleration equations mentioned above. These simplified equations assume no outside resistances, like friction, and allow us to focus solely on the movement caused by the provided force.
- Kinematics assumes motion in a defined space usually represented on a set of axes, making it easier to apply mathematical formulas.
- It also provides visual insight into the object's path through position-time and velocity-time graphs, useful for data interpretation and prediction.
force and motion
Newton's Second Law, \( F = ma \), is essential here as it forms the backbone for predicting how an object reacts to a given force. In the exercise, the applied force initiates the puck's motion on the frictionless rink, providing a straightforward example of this relationship.
- This law tells us that the acceleration of an object is directly proportional to the force applied, given a constant mass, meaning more force means more acceleration.
- Motion depends significantly on the nature and direction of the applied force.
Understanding force and motion is fundamental to mastering how various systems work, from sports dynamics to complex real-world machinery operations. Grasping these principles allows students to predict and analyze motion resulting from various forces effectively.