A 100-kg boulder moving at 1.2 m/s at a 30-degree angle hits a stationary 300-kg mine car. After the impact, they become coupled. The car was rolling at 0.6 m/s towards the impact. What is the final velocity of the coupled system?
Explanation
This problem combines the principles of conservation of linear momentum for a multi-body system in two dimensions. The total momentum vector of the system (cars and boulder) before the final state is equal to the total momentum vector after coupling. The solution requires breaking down the initial velocities into components and summing them before solving for the final velocity.
Other questions
According to the generalized Newton's second law for a system of particles, the resultant of the external forces on the system equals the total mass of the system times what quantity?
For a system of particles, the sum of all internal forces is identically zero. Why is this the case?
The total kinetic energy of a mass system can be expressed as the sum of two components. What are these two components?
What is the linear momentum, G, of a system of particles with constant mass 'm' and mass-center velocity 'v_bar'?
In the analysis of steady mass flow through a rigid container, what does the term 'm_dot * Δv' in the equation ΣF = m_dot * Δv represent?
For a system with variable mass, such as a rocket, what does the term 'm_dot * u' in the equation of motion ΣF = m * v_dot + m_dot * u represent?
Under what condition is the total linear momentum of a system of particles conserved?
Under what condition is the total angular momentum of a system of particles about its mass center, G, conserved?
A system of three particles with masses m, 2m, and 4m has a single external force F applied to the 4m particle in the positive y-direction. The system rests on a smooth horizontal surface. What is the acceleration of the system's mass center?
In the analysis of steady mass flow, the force-momentum equation is ΣF = m_dot(v_out - v_in). What does the term ΣF represent in this context?
A jet aircraft with a mass of 4.6 Mg has a drag of 32 kN at a speed of 1000 km/h. It consumes air at 106 kg/s and fuel at 4 kg/s, with an exhaust velocity of 680 m/s relative to the nozzle. To maintain a constant speed of 1000 km/h, what is the maximum angle of elevation at which the jet can fly?
A jet ski has reached its maximum velocity of 70 km/h. Its pump discharges salt water (density 1025 kg/m^3) from a horizontal 50-mm diameter nozzle at a rate of 0.082 m^3/s. The water enters the intake at the ski's velocity. What is the resistance R of the water to the hull at this speed?
An empty 50-kg tank is propelled by a force P while scooping up fresh water from a stream flowing at 1.5 m/s in the opposite direction. At an instant when the tank contains 80 kg of water and is moving at 2 m/s with an acceleration of 0.4 m/s^2, what is the required force P? The scoop entrance area is 2000 mm^2.
A rocket with an initial total mass m0 is fired vertically. It burns fuel at a constant rate, and the exhaust gas has a constant relative velocity u. If the residual mass is mb, what is the maximum velocity reached by the rocket, neglecting air resistance?
The end of a chain of length L and mass per unit length rho is lifted vertically from a piled platform with a constant velocity v. What is the force P required as a function of the height x of the end above the platform?
A 450-kg ram of a pile driver falls 1.4 m from rest and strikes a 240-kg pile. The ram and pile move together with no rebound. What is the velocity of the pile and ram immediately after impact?
An 8-Mg truck is at rest on the deck of a 240-Mg barge, which is also at rest in still water. If the truck starts and drives toward the bow at a speed of 6 km/h relative to the barge, what is the resulting speed of the barge?
What is the primary reason for analyzing the angular momentum of a system about the mass center G (HG) rather than just a fixed point O (HO)?
A fire tug discharges salt water (density 1030 kg/m^3) with a nozzle velocity of 40 m/s at a rate of 0.080 m^3/s. The discharge is angled 30 degrees above the horizontal. To maintain a fixed position, what propeller thrust T must the tug develop?
For the generalized Newton's second law ΣF = ma to be valid, what is a key requirement for the reference axes from which the acceleration 'a' of the mass center is measured?
When is it permissible to use the simplified force-momentum equation ΣF = d(mv)/dt for a system whose mass is changing?
A 10-kg block is at rest on a rough incline (angle 20 degrees, static friction 0.50, kinetic friction 0.40). At t=0, a time-varying force P, directed up the incline, is applied. P increases linearly from 0 to 100 N over 6 seconds. What is the velocity of the block at t = 3 seconds?
Two freight cars, A (130,000 lb) and C (150,000 lb), are rolling at 2 mi/hr and 1.5 mi/hr respectively. They are uncoupled. What is the magnitude of the total linear momentum of this two-car system?
Which principle is most directly applicable for analyzing the velocity change of a system over a specified time interval when forces are known as functions of time?
A 12-Mg truck drives onto a 350-Mg barge at 20 km/h and brakes to a stop on the deck. The barge is initially at rest in calm water. What is the final speed of the barge?
Why is the term 'm_dot * v' often referred to as 'momentum flux' in the context of steady mass flow?
In the variable-mass equation of motion, ΣF = m*dv/dt + m_dot*u, what does the variable 'u' represent?
Three identical spheres are welded to a light rigid frame rotating in a horizontal plane about a vertical axis through O. A couple M_O = 30 N-m is applied for 5 seconds to the frame, which is initially rotating at 20 rad/s. What is the new angular velocity? Each sphere has a mass of 4 kg and is at a radius of 0.5 m, another at 0.4 m, and the last at 0.6 m.
The angular momentum of a system of particles about an arbitrary, possibly accelerating, point P is given by HP = HG + rho_bar x (m*v_bar), where HG is the angular momentum about the mass center G. What does the term rho_bar x (m*v_bar) represent?
A two-stage rocket is in space when the first stage burns out and the 1400 kg second stage ignites. The second stage burns fuel at 5.2 kg/s with an exhaust velocity of 3000 m/s relative to its nozzle. What is the acceleration of the second stage 60 seconds after ignition, assuming a constant g of 8.70 m/s^2?
A rotary snow plow moves at a constant 20 km/h and discharges 60 Mg of snow per minute with a velocity of 12 m/s relative to the plow, at an angle of 45 degrees. What is the tractive force P on the tires required to maintain the constant speed?
If a system is conservative and no external forces (other than potential forces like gravity) do work on it, which quantity is conserved?
The total linear momentum of a system of five particles at time t = 2.2 s is G = 3.4i - 2.6j + 4.6k kg-m/s. At t = 2.4 s, the momentum is G = 3.7i - 2.2j + 4.9k kg-m/s. What is the magnitude of the time average of the resultant external force on the system during this interval?
A 4-kg cart is at rest. It is acted on by a horizontal force that is initially 20 N and increases linearly to 30 N over 4 seconds. Neglecting friction, what is the velocity of the cart at t = 3 seconds?
A 1000-kg car is moving at 30 km/h. It accelerates uniformly to 70 km/h over 15 seconds. The average rolling resistance is 500 N. What is the average tension in the tow cable pulling the car?
A 1.4 Mg test vehicle is accelerated from rest by a high-velocity water jet (v = 150 m/s) from a 140-mm-diameter nozzle. Frictional resistance is 10 percent of the vehicle's weight. What is the velocity of the vehicle 3 seconds after release?
An ideal rope of length L and mass per unit length rho is pulled by a force P, giving end A a constant velocity v. The rope is doubled back on itself, with end B fixed. What is the force P required?
The principle ΣF = G_dot (where G is linear momentum) is an alternative form of which fundamental law?
A 0.3 kg block is coated with a weak adhesive and pressed onto a turntable. The turntable starts from rest at t=0 and accelerates uniformly with alpha = 2 rad/s^2. If the adhesive fails at t = 3 s, what was the ultimate shear force supported by the adhesive? The block is 0.5 m from the center.
A system consists of two 2-kg balls on a smooth horizontal surface. They are initially at rest. A vertical force of 60 N is applied to the junction of the wires connecting them. What is the initial vertical acceleration of each ball?
The work-energy equation for a system of particles, U_1-2 = ΔT, relates the total work done to the change in total kinetic energy. What forces are included in the work term U_1-2?
A 1500-kg car has a velocity of 30 km/h up a 10-percent grade. The driver applies power for 8 seconds to reach 60 km/h. What is the time average of the total tangential force F exerted on the tires during this period?
An empty 25 Mg coal car is moving at 1.2 m/s when a hopper begins releasing coal into it at a rate of 4 Mg per second. How far has the car moved after 32 Mg of coal have been deposited?
A 1200-kg car, initially at rest, is towed by a cable. The car accelerates uniformly from 30 km/h to 70 km/h over a 15-s interval. What is the average power transmitted by the tow cable during this interval, assuming the cable is horizontal and rolling resistance is a constant 500 N?
An open-link chain of length L and mass rho per unit length is released from rest with a small overhang x to initiate motion over a smooth corner. What is the velocity of the chain when the last link A reaches the corner?
A jet of water is deflected by a fixed 60-degree vane. If fresh water (1000 kg/m^3) issues from a nozzle at 30 m/s at a rate of 0.05 m^3/s and is split into two equal streams, what is the force F required to hold the vane?
A 100-lb block is initially stationary on a surface where the static friction coefficient is 0.60 and the kinetic friction is 0.40. A force P, which increases linearly from 0 to 80 lb over 4 seconds, is applied horizontally. What is the speed of the block at t = 4 seconds?
An 88-kg sprinter reaches a maximum speed of 11 m/s in 2.5 s with uniform acceleration, starting from rest. What is his power output at the instant his speed is 5 m/s?
A 900-kg motorized unit A is used to slow the descent of a 600-kg bucket B from 3 m/s to 0.5 m/s over 6 seconds. What is the average force R which the cable system exerts to support unit A during this time?