Work and Energy: A Second Way to Solve Motion Problems

1) Work: How Forces Transfer Energy Along a Displacement

So far, many motion problems have been solved by writing forces, applying ΣF = ma, and then using kinematics to connect acceleration to velocity and position. The energy method is a second pathway: instead of tracking acceleration at every moment, you track how forces change the object’s energy between two positions.

Work as “force in the direction of motion”

Work is a number that measures how much a force helps or opposes motion along a displacement. For a constant force over a straight displacement, the work done by a force is

W = F d cos(θ)

where θ is the angle between the force vector and the displacement vector.

• If the force points in the same direction as the displacement (θ = 0), then cos(θ) = 1 and work is positive: the force adds energy.
• If the force points opposite the displacement (θ = 180°), then cos(θ) = -1 and work is negative: the force removes energy.
• If the force is perpendicular to the displacement (θ = 90°), then cos(θ) = 0 and the work is zero: the force changes direction but not speed (common in ideal circular motion).

Dot product logic (beginner-friendly)

The formula W = F d cos(θ) is the “dot product” idea in plain terms: only the component of the force along the displacement matters. You can rewrite it as

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W = (F_parallel) d

where F_parallel = F cos(θ) is the part of the force pointing along the motion. This is often easier than thinking about angles.

Work by multiple forces

If several forces act, each force can do work. The total work is the sum:

W_total = W1 + W2 + W3 + ...

Important practical note: some forces often do zero work because they are perpendicular to the displacement. For example, the normal force from a surface is often perpendicular to the motion along the surface, so it does no work in many ramp problems (as long as the surface does not move and there is no penetration).

Units check

Work has units of N·m, called a joule (J). A joule is the amount of work done by a 1 N force acting over 1 m in the direction of motion.

Step-by-step: computing work in typical situations

• Constant force, straight path: find F_parallel, then multiply by d.
• Force changes with position (like a spring): break into small pieces and add them (this becomes an integral in advanced form). For a spring, you can use a known result (introduced below).
• Curved path: work depends on the component of force along the instantaneous direction of motion; for conservative forces you can often avoid the path details by using potential energy.

2) Work–Energy Theorem: Connecting Work to Speed Without Time

Kinetic energy

Kinetic energy is the energy of motion:

K = (1/2) m v^2

It depends on speed v, not direction.

The work–energy theorem

The key bridge between forces and motion is:

W_total = ΔK = K_f - K_i

In words: the net work done by all forces equals the change in kinetic energy.

Why this can bypass time and intermediate details

Many problems ask for a speed at a certain position, or a stopping distance, without caring about how long it took. The work–energy theorem directly connects forces over distance to changes in speed, often without solving for acceleration as a function of time.

This is especially helpful when:

• Acceleration is not constant.
• Forces vary with position (springs, varying gravity in advanced cases).
• You only care about initial and final speeds/positions.

Step-by-step template (work–energy)

1. Choose the object (or system) you are analyzing.
2. Identify the initial state (position, speed) and final state.
3. Compute work done by each external force over the displacement, then sum to get W_total.
4. Set W_total = (1/2)m v_f^2 - (1/2)m v_i^2 and solve for the unknown.

Example: constant push on a cart

A cart of mass m starts from rest and is pushed with a constant horizontal force F over a distance d on a frictionless floor. Find its final speed.

• Work by push: W = F d
• Normal and weight do zero work (perpendicular to motion).
• W_total = F d = ΔK = (1/2)m v_f^2 - 0
• v_f = sqrt(2 F d / m)

No time variable appears, and you did not need to compute acceleration first.

3) Conservative Forces, Potential Energy, and Mechanical Energy

Conservative force idea

A force is called conservative if the work it does depends only on the starting and ending positions, not on the path taken. For conservative forces, it is convenient to store their effect in a potential energy function.

Two common conservative forces in beginner mechanics are:

• Gravity near Earth’s surface (approximately constant g).
• Ideal spring force (Hooke’s law).

Potential energy and the work done by conservative forces

For a conservative force, the work done by that force is related to potential energy by

W_conservative = -ΔU

This means: if potential energy decreases, the force does positive work (it speeds you up); if potential energy increases, the force does negative work (it slows you down unless something else adds energy).

Gravitational potential energy near Earth

Choose a vertical coordinate y (upward positive). Near Earth, gravitational potential energy is

U_g = m g y

Only differences matter: ΔU_g = m g (y_f - y_i). You are free to choose where U_g = 0 (the “zero reference”). The physics does not change as long as you are consistent.

Spring potential energy

For an ideal spring with constant k, and displacement x measured from the spring’s relaxed length, the spring potential energy is

U_s = (1/2) k x^2

This comes from the fact that the spring force grows with displacement, so the work is not simply F d with a single constant F.

Mechanical energy

Mechanical energy is the sum of kinetic and potential energies:

E_mech = K + U

where U may include multiple potential energies (gravity + spring, etc.).

Conservation of mechanical energy (when it applies)

If only conservative forces do work (or if nonconservative forces do zero net work), then mechanical energy is conserved:

K_i + U_i = K_f + U_f

This is often the fastest way to connect speeds and heights, or speeds and spring compressions, without computing forces along the path.

Step-by-step template (conservation of mechanical energy)

1. Decide what is included in your system (object alone, or object + Earth, or object + spring, etc.).
2. List the initial and final values of K and each relevant U.
3. Check whether nonconservative forces (like kinetic friction) do work. If not, use K_i + U_i = K_f + U_f.
4. Choose a convenient zero reference for potential energy (often the lowest point for gravity, or the relaxed spring length for springs).

Example: speed at the bottom of a ramp (no friction)

A block starts from rest at height h above the bottom of a frictionless ramp. Find its speed at the bottom.

• Choose bottom as U_g = 0.
• Initial: K_i = 0, U_i = mgh
• Final: K_f = (1/2) m v^2, U_f = 0
• Conservation: mgh = (1/2) m v^2
• v = sqrt(2gh)

Notice the ramp angle and length do not matter (in the ideal frictionless case) because gravity is conservative.

4) Friction and Nonconservative Work: Where the Energy Goes

Nonconservative forces

Forces like kinetic friction, air drag, and many applied forces (depending on how they vary) are nonconservative: the work they do depends on the path and typically converts mechanical energy into other forms (thermal/internal energy, sound, deformation).

Energy accounting with nonconservative work

A useful general statement is:

ΔE_mech = W_nonconservative

Equivalently:

K_i + U_i + W_nonconservative = K_f + U_f

If W_nonconservative is negative (as with kinetic friction), mechanical energy decreases. That “missing” mechanical energy is not destroyed; it becomes internal energy (heating of surfaces, microscopic deformation, etc.).

Work done by kinetic friction (common model)

If kinetic friction has magnitude f_k and acts opposite the motion along a path length d, then

W_friction = - f_k d

On a horizontal surface, f_k = &mu_k N = &mu_k m g. On an incline, N changes, so compute N from the geometry before using f_k = &mu_k N.

Choosing system boundaries (a practical rule)

• If you include Earth in the system, then gravity becomes an internal conservative interaction and you can use U_g.
• If you include the spring in the system, then the spring force becomes internal conservative and you can use U_s.
• Friction is usually treated as a nonconservative interaction that transfers energy to internal/thermal energy; you represent it via W_friction (or by explicitly tracking thermal energy if desired).

Zero reference consistency for potential energy

Potential energy has an arbitrary zero. Pick a reference that simplifies the algebra (often the final point, the lowest point, or the relaxed spring length). Only differences ΔU affect the physics, so any consistent choice works.

Structured Problem Sets (with Solution Outlines)

A) Ramps (gravity, normal force, possibly friction)

A1. Frictionless ramp speed A block starts from rest at vertical height h = 2.0 m. Find speed at the bottom.

• Use mgh = (1/2) m v^2.
• v = sqrt(2gh).

A2. Ramp with kinetic friction A block of mass m slides down a ramp of length L at angle α with coefficient &mu_k, starting from rest. Find speed at the bottom.

• Choose system: block + Earth (so use U_g).
• Energy equation: K_i + U_i + W_f = K_f + U_f.
• U_i - U_f = mg(L sinα) (vertical drop is L sinα).
• N = mg cosα so f_k = &mu_k mg cosα.
• W_f = - f_k L = -&mu_k mg cosα L.
• Compute: mgL sinα - &mu_k mg cosα L = (1/2) m v^2.
• v = sqrt(2gL(sinα - &mu_k cosα)) (requires sinα > &mu_k cosα to actually speed up).

A3. “How far up does it go?” A block is launched up a frictionless incline with initial speed v0. Find the maximum vertical rise h.

• At the top, v = 0.
• (1/2) m v0^2 = mgh so h = v0^2/(2g).

B) Springs (variable force, spring potential)

B1. Spring launch on a frictionless track A cart of mass m is pushed against a spring (k) compressing it by x, then released on a frictionless horizontal track. Find the cart’s speed when the spring returns to its relaxed length.

• Choose system: cart + spring (so use U_s).
• Initial: K_i = 0, U_i = (1/2)kx^2.
• Final at relaxed length: U_f = 0, K_f = (1/2)mv^2.
• (1/2)kx^2 = (1/2)mv^2 so v = x sqrt(k/m).

B2. Spring + gravity (vertical motion) A mass m is attached to a vertical spring (k). It is released from rest at a position where the spring is stretched by x_i (measured from relaxed length). Find speed when it passes a position x_f.

• Choose coordinate and a consistent zero for gravitational potential (e.g., set U_g = 0 at x = 0 or at one of the positions).
• Use K_i + U_s(x_i) + U_g(y_i) = K_f + U_s(x_f) + U_g(y_f).
• Be careful: in a vertical spring, x and y are linked by geometry (often y = -x depending on sign convention).

B3. Spring with friction on a horizontal surface A block slides on a rough horizontal surface into a spring and compresses it by x_max before stopping. Given v0, &mu_k, k, find x_max.

• Initial: K_i = (1/2)mv0^2, U_s,i = 0.
• Final at max compression: K_f = 0, U_s,f = (1/2)k x_max^2.
• Friction work over distance x_max: W_f = -&mu_k mg x_max.
• Energy: (1/2)mv0^2 - &mu_k mg x_max = (1/2)k x_max^2.
• Solve the quadratic for x_max (choose the positive root).

C) Braking distance (energy method in everyday motion)

C1. Constant braking force A car of mass m moving at speed v0 experiences a constant braking force of magnitude F_b opposite its motion. Find stopping distance d.

• Work by brakes: W = -F_b d.
• ΔK = 0 - (1/2) m v0^2.
• -F_b d = -(1/2) m v0^2 so d = (m v0^2)/(2F_b).

C2. Braking limited by tire-road friction If maximum braking is limited by kinetic friction with coefficient &mu_k on level ground, estimate stopping distance.

• Max friction force: f_k = &mu_k mg.
• Work: -&mu_k mg d = -(1/2) m v0^2.
• d = v0^2/(2 &mu_k g).

C3. Braking on a downhill slope A vehicle goes downhill a distance d on a slope angle α while braking with constant magnitude F_b. Find the speed change.

• Work by gravity along slope: W_g = +mg sinα d.
• Work by brakes: W_b = -F_b d.
• Total work: W_total = (mg sinα - F_b)d = ΔK.
• Use (1/2)m(v_f^2 - v_i^2) = (mg sinα - F_b)d.

D) Mixed force fields (gravity + spring + friction + applied work)

D1. Ramp into a spring with friction on the ramp A block starts from rest at height h, slides down a rough ramp (length L, angle α, coefficient &mu_k), then compresses a spring at the bottom (spring constant k) on a frictionless horizontal section. Find maximum compression x_max.

• Choose system: block + Earth + spring (so use U_g and U_s).
• Initial: K_i=0, U_g,i = mgh, U_s,i=0.
• Nonconservative work: friction only on ramp: W_f = -&mu_k (mg cosα) L.
• Final at max compression: K_f=0, U_g,f=0 (choose bottom as zero), U_s,f=(1/2)k x_max^2.
• Energy: mgh + W_f = (1/2)k x_max^2.
• Solve: x_max = sqrt((2(mgh - &mu_k mg cosα L))/k) (requires the expression inside the square root to be positive).

D2. Applied force over part of the path A person pulls a sled with a rope at angle θ above horizontal with constant tension T over distance d on level ground with kinetic friction &mu_k. Find the final speed given initial speed v_i.

• Work by tension: W_T = (T cosθ) d (only horizontal component contributes to work along horizontal displacement).
• Normal force: N = mg - T sinθ (reduced by upward rope component).
• Friction: f_k = &mu_k N, work W_f = -f_k d = -&mu_k (mg - T sinθ) d.
• Total work: W_total = W_T + W_f.
• Work–energy: (1/2)m(v_f^2 - v_i^2) = (T cosθ) d - &mu_k (mg - T sinθ) d.

D3. Choosing the system boundary (concept check) For each scenario, decide whether it is most convenient to include Earth and/or the spring in the system, and whether you will use U or explicit work:

Problem-solving checklist (energy method)

• Write the energy equation first: K_i + U_i + W_nc = K_f + U_f.
• Pick a system: include Earth/spring if you want to use U_g/U_s.
• Choose zero references: set U_g = 0 at a convenient height; set U_s = 0 at relaxed length.
• Compute only what matters: forces perpendicular to displacement do zero work; conservative forces can be handled via U.
• Check signs: friction work is negative; gravity work depends on whether you go down or up.
• Sanity-check: if friction is present, mechanical energy should decrease unless an external agent adds energy.