A tank truck for washing down streets has a total weight of...

A tank truck for washing down streets has a total weight of 20,000 lb when its tank is full. With the spray turned on, 80 lb of water per second issue from the nozzle with a velocity of 60 ft/sec relative to the truck at the $30^{\circ}$ angle shown. If the truck is to accelerate at the rate of $2 \mathrm{ft} / \mathrm{sec}^{2}$ when starting on a level road, determine the required tractive force $P$ between the tires and the road when $(a)$ the spray is turned on and ( $b$ ) the spray is turned off.

Key Concepts

Newton's Second Law

Newton's Second Law establishes the relationship between force, mass, and acceleration, indicating that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In problems involving vehicles accelerating from rest, this law is used to calculate the tractive force required to achieve a desired acceleration, taking into account the mass of the system involved.

Mass Flow Rate

Mass flow rate refers to the amount of mass passing through a given cross-section per unit time. In the context of a fluid jet exiting a nozzle, this concept is crucial as it quantifies how much mass is being ejected, thereby impacting the momentum change and resulting reaction force acting on the vehicle.

Momentum Conservation in Fluid Jets

The principle of momentum conservation is applied to situations where mass is ejected from a system, such as water being sprayed from a truck. The loss of momentum caused by the discharged water results in a reactive force (or thrust) that can oppose the truck's motion. This concept is important when considering additional forces that affect the overall dynamics of the vehicle.

Relative Velocity

Relative velocity is the speed of one object as observed from another moving object. When the water is ejected from the truck at a given velocity relative to it, determining the actual momentum change requires analyzing the water's speed from the truck's frame of reference, which influences the magnitude of the resulting reactive force.

Force Analysis in Accelerating Systems

Force analysis in accelerating systems involves summing all external forces acting on the system to determine the net force required for a given motion. In this problem, it is necessary not only to account for the force needed to accelerate the mass of the truck but also to include the reactive forces produced by the expelled water, providing a comprehensive view of all forces in play.

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