George left the lights on in his truck while at a truck stop...

George left the lights on in his truck while at a truck stop in Kansas and his battery went dead. Fortunately, his friend $\mathrm{Al}$ is there, although $\mathrm{Al}$ is driving his Geo Metro. Since the road is very flat, George is able to convince $\mathrm{Al}$ to give his truck a long, slow push to get it up to 20 miles/hour. At this speed, George can engage the truck's clutch and the truck's engine should start up. (See Fig. 6-89.) (a) Al begins to push the truck. It takes him 5 minutes to get the truck up to a speed of 20 miles/hour. Draw separate free-body diagrams for the Geo and for the truck during the time that Al's Geo is pushing the truck. List all the horizontal forces in order by $\mathrm{mag}$ nitude from largest to smallest. If any are equal, state that explicitly. Explain your reasoning. (b) If the truck is accelerating uniformly over the 5 minutes, how far does Al have to push the truck before George can engage the clutch? (c) Suppose the mass of the truck is $4000 \mathrm{~kg}$, the mass of the car is $800 \mathrm{~kg}$, and the coefficient of static friction between the vehicles and the road is $0.1$. At one instant when they are trying to get the truck moving, the car is pushing the truck and exerting a force of $1000 \mathrm{~N}$, but neither vehicle moves. What is the static frictional force between the truck and the road? Explain your reasoning.

George left the lights on in his truck while at a truck stop in Kansas and his battery went dead. Fortunately, his friend $\mathrm{Al}$ is there, although $\mathrm{Al}$ is driving his Geo Metro. Since the road is very flat, George is able to convince $\mathrm{Al}$ to give his truck a long, slow push to get it up to 20 miles/hour. At this speed, George can engage the truck's clutch and the truck's engine should start up. (See Fig. 6-89.)
(a) Al begins to push the truck. It takes him 5 minutes to get the truck up to a speed of 20 miles/hour. Draw separate free-body diagrams for the Geo and for the truck during the time that Al's Geo is pushing the truck. List all the horizontal forces in order by $\mathrm{mag}$ nitude from largest to smallest. If any are equal, state that explicitly. Explain your reasoning.
(b) If the truck is accelerating uniformly over the 5 minutes, how far does Al have to push the truck before George can engage the clutch?
(c) Suppose the mass of the truck is $4000 \mathrm{~kg}$, the mass of the car is $800 \mathrm{~kg}$, and the coefficient of static friction between the vehicles and the road is $0.1$. At one instant when they are trying to get the truck moving, the car is pushing the truck and exerting a force of $1000 \mathrm{~N}$, but neither vehicle moves. What is the static frictional force between the truck and the road? Explain your reasoning.

Key Concepts

Free-Body Diagrams

A free-body diagram is a visual tool used to identify and represent all the forces acting on an object. In problems involving vehicles being pushed, it helps in isolating each object (the truck and the car) to clearly view forces like applied push, friction, and normal force, allowing for a systematic breakdown of the dynamics at play.

Net Force Analysis

Net force analysis involves summing all the horizontal forces acting on an object to determine the unbalanced force that causes acceleration. This method is essential when comparing forces of different magnitudes, such as the push provided by one vehicle versus frictional forces, to understand why and how the motion of the truck begins or remains static.

Static Friction

Static friction is the force that prevents an object from starting to move when a force is applied. It plays a critical role in determining the threshold force required to initiate motion. Understanding static friction is important in scenarios where vehicles must overcome the resistance between their tires and the road before they can start moving.

Newton's Laws of Motion

These laws describe the relationship between force and motion. They are the foundation for analyzing how forces acting on a vehicle, such as the push from another vehicle and frictional resistance, result in acceleration or equilibrium. Newton’s third law, for example, explains that every force has an equal and opposite reaction, which is key to understanding the interaction between the two vehicles.

Uniform Acceleration Kinematics

Uniform acceleration kinematics uses equations that relate displacement, time, velocity, and acceleration under constant acceleration conditions. This concept is crucial when calculating how far an object travels as it accelerates uniformly from rest to a given speed, which is directly applicable to determining the distance the truck covers while being pushed.

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