Problem 58 - University Physics with Modern Physics

Question

A truck with mass \(m\) has a brake failure while going down an icy mountain road of constant downward slope angle \(\alpha\) (\(\textbf{Fig. P7.58}\)). Initially the truck is moving downhill at speed \(v_0\). After careening downhill a distance \(L\) with negligible friction, the truck driver steers the runaway vehicle onto a runaway truck ramp of constant upward slope angle \(\beta\). The truck ramp has a soft sand surface for which the coefficient of rolling friction is \(\mu_r\). What is the distance that the truck moves up the ramp before coming to a halt? Solve by energy methods.

Step-by-Step Solution

Answer

The distance up the ramp before stopping is \(d = \frac{v_0^2 + 2gL\sin(\alpha)}{2g(\sin(\beta) + \mu_r \cos(\beta))} \).

Step 1: Analyze Initial Energy

When the truck starts, it possesses both kinetic energy due to its speed and potential energy due to its height on the slope. The initial kinetic energy is given by \( KE_i = \frac{1}{2}mv_0^2 \). Since we're using the change in height as the reference for zero potential energy, initial potential energy when the truck starts moving down the slope will depend on its vertical height change, \( PE_i = mgL\sin(\alpha) \), where \(L\sin(\alpha)\) is the vertical component of the downhill distance.

Step 2: Analyze Energy As Truck Transitions To Ramp

As the truck reaches the base of the ramp, we calculate its speed using energy conservation on the slope. The truck has lost potential energy \(mgL\sin(\alpha)\) and maintains the kinetic energy from initial speed and loss in potential energy at the base of the ramp, calculated as \( KE_{base} = \frac{1}{2}mv_{base}^2 = \frac{1}{2}mv_0^2 + mgL\sin(\alpha) \). Using \(v_{base}\) find that \( v_{base}^2 = v_0^2 + 2gL\sin(\alpha) \).

Step 3: Construct Energy Balance on Ramp

This balance involves accounting for the kinetic energy at the base and subtracting energy lost to friction and potential energy gained on the ramp. The ramp's vertical component affects potential energy: \( PE_{final} = mgd\sin(\beta) \). Work done against friction, \( W_f = \mu_r mgd\cos(\beta) \), reduces energy. By energy conservation: \[ \frac{1}{2}mv_{base}^2 = mgd\sin(\beta) + \mu_r mgd\cos(\beta) \].

Step 4: Solve for Distance on Ramp

Solve the equation for distance \(d\). Start with cancelling \(m\) from the energy equation: \[ \frac{1}{2}v_{base}^2 = gd\sin(\beta) + \mu_r gd\cos(\beta) \]. Insert \(v_{base}^2 = v_0^2 + 2gL\sin(\alpha) \) from earlier, \[ \frac{1}{2}(v_0^2 + 2gL\sin(\alpha)) = gd(\sin(\beta) + \mu_r \cos(\beta)) \]. Divide and rearrange to solve for \(d\): \[ d = \frac{v_0^2 + 2gL\sin(\alpha)}{2g(\sin(\beta) + \mu_r \cos(\beta))} \].

Key Concepts

Kinetic Energy

In the scenario of the truck barreling down the mountain road, kinetic energy plays a significant role. Kinetic energy is essentially the energy an object possesses due to its motion, and it increases with both mass and speed.

For the truck, this initial kinetic energy (\( KE_i \)) is calculated using the formula \( KE_i = \frac{1}{2}mv_0^2 \), where \( m \) represents the mass of the truck and \( v_0 \) is its initial speed. This energy is what propels the truck forward on the icy slope.

• Kinetic energy depends on both the mass and the square of velocity.
• Even a slight increase in velocity results in a significant increase in kinetic energy.
• Kinetic energy will transform as the truck's speed changes.

As the truck moves, this kinetic energy changes due to various forces it encounters, forming a crucial part of understanding energy conservation.

Potential Energy

Potential energy refers to the stored energy of an object due to its position or height. For our truck scenario, potential energy (\( PE \)) is defined relative to the height on the slope.

Initially, the truck's potential energy can be given by the expression \( PE_i = mgL\sin(\alpha) \), where \( g \) is the acceleration due to gravity, \( L \) is the distance down the slope, and \( \sin(\alpha) \) signifies the slope's angle contributing to vertical height.

• Potential energy is maximal when the truck is at the highest initial point.
• The formula integrates gravitational force, highlighting the impact of gravity on energy conservation.
• The truck loses potential energy, which transforms primarily into kinetic energy as it descends.

This interplay of energies is vital for understanding how vehicles convert stored energy to motion energy, affecting how fast they travel downhill.

Rolling Friction

Rolling friction comes into play particularly when considering the truck's attempt to stop on the upward-sloping ramp. This type of friction resists the motion of a rolling object across a surface.

As the truck ascends the slope, rolling friction contributes to the energy loss that must be overcome for the truck to come to a halt. It opposes the truck’s motion up the ramp and is quantified using the coefficient of rolling friction \( \mu_r \) and the truck’s weight component aligned with the ramp’s surface: \( W_f = \mu_r mgd\cos(\beta) \).

• Rolling friction is generally less significant than static or kinetic friction, but still critical in vehicle dynamics.
• The angle \( \beta \) affects how gravity and rolling friction interact.
• This resistance is key in energy calculations, influencing how far the truck can travel up the slope.

Understanding rolling friction is essential in predicting motion behavior, key for both everyday transport and energy conservation physics.