Problem 82 - University Physics with Modern Physics

Question

You are riding your motorcycle one day down a wet street that slopes downward at an angle of \(20^{\circ}\) below the horizontal. As you start to ride down the hill, you notice a construction crew has dug a deep hole in the street at the bottom of the hill. A siberian tiger, escaped from the City Zoo, has taken up residence in the hole. You apply the brakes and lock your wheels at the top of the hill, where you are moving with a speed of 20 \(\mathrm{m} / \mathrm{s} .\) The inclined street in front of you is 40 \(\mathrm{m}\) long. (a) Will you plunge into the hole and become the tiger's lunch, or do you skid to a stop before you reach the hole? (The coefficients of friction between your motorcycle tires and the wet pavement are \(\mu_{\mathrm{s}}=0.90\) and \(\mu_{\mathrm{k}}=0.70 .\) ) ( b ) What must your initial speed be if you are to stop just before reaching the hole?

Step-by-Step Solution

Answer

(a) The motorcycle will stop before reaching the hole if the calculated stopping distance is less than 40 m. (b) Calculate the initial speed using the kinematic equation for a 40 m descent.

Step 1: Understanding the Problem

We are given a motorcycle moving down a slope at an angle \(20^{\circ}\) with an initial speed of 20 m/s. The slope is 40 m long, and we need to determine if the motorcycle stops before reaching the hole. The coefficients of static and kinetic friction are \(\mu_s = 0.90\) and \(\mu_k = 0.70\), respectively. We must calculate the stopping distance using these values.

Step 2: Calculate the Forces Involved

First, calculate the gravitational force component acting down the slope (parallel to the surface): \(F_{\text{gravity}} = mg \sin \theta\). The normal force is \(F_{\text{normal}} = mg \cos \theta\). The frictional force (kinetic) is \(F_{\text{friction}} = \mu_k mg \cos \theta\).

Step 3: Set Up the Motion Equation

According to Newton's second law, the net force acting on the motorcycle is the difference between the gravitational force component along the incline and the frictional force: \(F_{\text{net}} = mg \sin \theta - \mu_k mg \cos \theta\). Substitute \(F_{\text{net}}\) into \(ma = F_{\text{net}}\) to find acceleration \(a = g \sin \theta - \mu_k g \cos \theta\).

Step 4: Calculate the Acceleration

Substituting values: \(a = 9.8 \sin 20^{\circ} - 0.70 \cdot 9.8 \cos 20^{\circ}\). Calculate \(a\) to find the actual deceleration experienced by the motorcycle.

Step 5: Determine Stopping Distance

Use the kinematic equation: \(v^2 = u^2 + 2as\). We know \(v = 0\) (final velocity), \(u = 20\, \text{m/s}\) (initial velocity), and \(s\) is the distance we want to calculate. Solve for \(s\) using the previously found \(a\).

Step 6: Compare Stopping Distance with Slope Length

Calculate \(s\) and compare it to the slope's length (40 m). If \(s \leq 40\, \text{m}\), the motorcycle stops in time. If it is greater, it does not.

Step 7: Solve for Minimum Initial Speed (Part b)

For the second part, use the same kinematics equation \(v^2 = u^2 + 2as\), where \(v = 0\), \(s = 40\,\text{m}\), and solve for \(u\) to find the maximum initial speed that allows the motorcycle to stop in exactly 40 m.

Key Concepts

Kinematics

Kinematics is a branch of physics that describes the motion of objects without considering the forces that cause this motion. In the given problem, we are primarily concerned with linear motion along an inclined plane, which is a common scenario in physics problems.

Key kinematic equations are used to relate the different variables involved in motion. These include the initial and final velocities, acceleration, time, and displacement. In this scenario, the problem provides the initial speed of the motorcycle and asks if it can stop before reaching a hole at the bottom of the slope.

To solve it, we can use the kinematic equation: \[v^2 = u^2 + 2as\] where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the displacement. By setting \(v = 0\), we can determine if the initial conditions allow the motorcycle to stop in the given distance.

Friction

Friction is the resistance that one surface or object encounters when moving over another. It plays a crucial role in the physics problem presented. The two types of friction involved here are static friction and kinetic friction, characterized by their respective coefficients, \(\mu_s\) and \(\mu_k\).

Static friction is what keeps the motorcycle from slipping when initially stationary, but once the wheels lock and the motorcycle begins sliding, kinetic friction takes over. In the calculation, kinetic friction is used because the motorcycle is skidding down the hill after the brakes are applied.

To determine the frictional force, we use the formula: \[F_{\text{friction}} = \mu_k F_{\text{normal}}\] where the normal force \(F_{\text{normal}}\) is the component of the gravitational force perpendicular to the incline. This frictional force provides the deceleration needed to stop the motorcycle.

Newton's Laws

Understanding Newton's Laws, especially the second law, is vital in analyzing the motion of the motorcycle. Newton's second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration: \[F = ma\] We apply this law to find the net force acting on the motorcycle as it moves down the incline. The net force is the difference between the downhill gravitational force component and the opposing frictional force.

Thus, we have: \[F_{\text{net}} = mg \sin \theta - \mu_k mg \cos \theta\] This net force determines the acceleration (or in this case deceleration) experienced by the motorcycle, which is essential to figure out if it will stop in time.

Inclined Planes

An inclined plane in physics refers to a flat surface tilted at an angle to the horizontal. This scenario provides a classic example of how the components of forces can be calculated using the angle of the incline.

In our exercise, the street is an inclined plane sloping downward at an angle of \(20^{\circ}\). The gravitational force can be broken into two components:

• The component parallel to the plane (\[mg \sin \theta\]) which drives the motorcycle downward.
• The component perpendicular to the plane (\[mg \cos \theta\]) providing the normal force which is essential for calculating friction.

These components allow us to simplify the complex interplay of forces acting on the motorcycle into understandable parts, facilitating the application of Newton's laws and kinematics to predict its motion or stopping capability.