When you need to add two or more vectors, use this step-by-step procedure:
? Select a coordinate system that is convenient. (Try to reduce the number of
components you need to find by choosing axes that line up with as many
vectors as possible.)
? Draw a labeled sketch of the vectors described in the problem.
? Find the x and y components of all vectors and the resultant components
(the algebraic sum of the components) in the x and y directions.
? If necessary, use the Pythagorean theorem to find the magnitude of the resultant vector and select a suitable trigonometric function to find the angle
that the resultant vector makes with the x axis.
? Select a coordinate system and resolve the initial velocity vector into x and y
? Follow the techniques for solving constant-velocity problems to analyze the
horizontal motion. Follow the techniques for solving constant-acceleration
problems to analyze the vertical motion. The x and y motions share the
same time of flight t.
APPLYING NEWTON’S LAWS
? Draw a simple, neat diagram of the system.
? Isolate the object whose motion is being analyzed. Draw a free-body diagram
for this object. For systems containing more than one object, draw separate
free-body diagrams for each object. Do not include in the free-body diagram
forces exerted by the object on its surroundings. Establish convenient coordinate
axes for each object and find the components of the forces along
? Apply Newton?s second law, F
ma, in component form. Check your dimensions
to make sure that all terms have units of force.
? Solve the component equations for the unknowns. Remember that you must
have as many independent equations as you have unknowns to obtain a
? Make sure your results are consistent with the free-body diagram. Also check
the predictions of your solutions for extreme values of the variables. By doing
so, you can often detect errors in your results.
CONSERVATION OF ENERGY
? Define your system, which may include two or more interacting particles, as
well as springs or other systems in which elastic potential energy can be
stored. Choose the initial and final points.
? Identify zero points for potential energy (both gravitational and spring). If
there is more than one conservative force, write an expression for the potential
energy associated with each force.
? Determine whether any nonconservative forces are present. Remember that
if friction or air resistance is present, mechanical energy is not conserved.
? If mechanical energy is conserved, you can write the total initial energy
at some point E(i)=PE(i)+KE(i). Then, write an expression for the total final energy at the final point that is of interest E(f)=PE(f)+KE(f). Because mechanical
energy is conserved, you can equate the two total energies and solve for the
quantity that is unknown.
? If frictional forces are present (and thus mechanical energy is not conserved),
first write expressions for the total initial and total final energies. In this
case, the difference between the total final mechanical energy and the total
initial mechanical energy equals the change in mechanical energy in the system
due to friction.
? Set up a coordinate system and define your velocities with respect to that system.It is usually convenient to have the x axis coincide with one of the initial
? In your sketch of the coordinate system, draw and label all velocity vectors
and include all the given information.
? Write expressions for the x and y components of the momentum of each object
before and after the collision. Remember to include the appropriate
signs for the components of the velocity vectors.
? Write expressions for the total momentum in the x direction before and after
the collision and equate the two. Repeat this procedure for the total momentum
in the y direction. These steps follow from the fact that, because
the momentum of the system is conserved in any collision, the total momentum
along any direction must also be constant. Remember, it is the momentum
of the system that is constant, not the momenta of the individual objects.
? If the collision is inelastic, kinetic energy is not conserved, and additional information is probably required. If the collision is perfectly inelastic, the final
velocities of the two objects are equal. Solve the momentum equations for
the unknown quantities.
? If the collision is elastic, kinetic energy is conserved, and you can equate the
total kinetic energy before the collision to the total kinetic energy after the
collision to get an additional relationship between the velocities.
OBJECTS IN STATIC EQUILIBRIUM
? Draw a simple, neat diagram of the system.
? Isolate the object being analyzed. Draw a free-body diagram and then show
and label all external forces acting on the object, indicating where those
forces are applied. Do not include forces exerted by the object on its surroundings.(For systems that contain more than one object, draw a separate
free-body diagram for each one.) Try to guess the correct direction for each
force. If the direction you select leads to a negative force, do not be
alarmed; this merely means that the direction of the force is the opposite of
what you guessed.
? Establish a convenient coordinate system for the object and find the components of the forces along the two axes. Then apply the first condition for
equilibrium. Remember to keep track of the signs of all force components.
? Choose a convenient axis for calculating the net torque on the object. Remember that the choice of origin for the torque equation is arbitrary; therefore,
choose an origin that simplifies your calculation as much as possible.
Note that a force that acts along a line passing through the point chosen as
the origin gives zero contribution to the torque and thus can be ignored.