Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 49a

Chapter 6, Problem 49a

Sam, whose mass is 75 kg, takes off across level snow on his jet-powered skis. The skis have a thrust of 200 N and a coefficient of kinetic friction on snow of 0.10. Unfortunately, the skis run out of fuel after only 10 s. What is Sam's top speed?

Verified step by step guidance

Step 1: Identify the forces acting on Sam. The forces include the thrust from the jet-powered skis (200 N), the kinetic friction force opposing motion, and the normal force balancing gravity. The friction force can be calculated using the formula: \( F_{friction} = \mu_k \cdot F_{normal} \), where \( \mu_k \) is the coefficient of kinetic friction and \( F_{normal} = m \cdot g \).

Step 2: Calculate the net force acting on Sam. The net force is given by \( F_{net} = F_{thrust} - F_{friction} \). Substitute the values for \( F_{thrust} \), \( \mu_k \), \( m \), and \( g \) to find \( F_{net} \).

Step 3: Use Newton's second law to find the acceleration. Newton's second law states \( F_{net} = m \cdot a \), where \( a \) is the acceleration. Rearrange the equation to solve for \( a \): \( a = \frac{F_{net}}{m} \). Substitute the values for \( F_{net} \) and \( m \) to calculate \( a \).

Step 4: Determine Sam's top speed using the kinematic equation \( v = u + a \cdot t \), where \( u \) is the initial velocity (0 m/s, since Sam starts from rest), \( a \) is the acceleration, and \( t \) is the time the skis are powered (10 s). Substitute the values for \( u \), \( a \), and \( t \) to find \( v \).

Step 5: Verify the units and ensure the calculations are consistent. The top speed \( v \) should be expressed in meters per second (m/s).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the formula F = ma, where F is the net force, m is mass, and a is acceleration. Understanding this law is crucial for calculating how forces like thrust and friction affect Sam's motion.

Friction

Friction is a force that opposes the relative motion of two surfaces in contact. The coefficient of kinetic friction quantifies this force, which can be calculated using the formula F_friction = μ_k * N, where μ_k is the coefficient of kinetic friction and N is the normal force. In this scenario, friction will affect Sam's acceleration and ultimately his top speed.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as velocity, acceleration, and time. To find Sam's top speed, we can use kinematic equations that relate these variables, particularly since we know the time he was in motion and the net acceleration.

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Sam, whose mass is 75 kg, takes off across level snow on his jet-powered skis. The skis have a thrust of 200 N and a coefficient of kinetic friction on snow of 0.10. Unfortunately, the skis run out of fuel after only 10 s. How far has Sam traveled when he finally coasts to a stop?