Engineering Mechanics: Dynamics 1 (Intuition + Application)

Learn one-dimensional particle kinematics for a vertical rocket, deriving displacement, velocity, and acceleration relationships, including instantaneous quantities, average values, and the no-forces assumption.

Solve a vertical rocket motion problem by deriving velocity as a function of position using definite integrals, applying initial conditions, and finding the velocity at 2000 meters.

Develop time as a function of position for vertical rocket motion by integrating 1/v with respect to position to reach 2000 meters, approximately 19.274 seconds.

An exercise in vertical motion compares bullet a at 0 s with 450 m/s and bullet b 3 s with 600 m/s under gravity to find when b overtakes a.

Analyze the vertical flight of two bullets fired at 450 and 600 m/s, with B launched 3 seconds after A, under gravity, to find when they align and the altitude.

Compute bullet B's position by integrating velocity from t=3 s, compare with bullet A, and determine their meet time around 10.27 s at about 4107 m, using constant-acceleration kinematic equations.

Explore bullet train acceleration using a time-acceleration graph, split into segments, from rest to positive acceleration, then deceleration to rest, and calculate total distance plus velocity and position graphs.

Compute velocity and position from train's acceleration by integrating from 0 to 30 seconds, yielding v(30)=45 m/s and s(30)=450 m. Then solve a quadratic to determine total time, 133.0 seconds.

Derive the time-based position and acceleration from a piecewise velocity function for a boat in one dimension, then compute the time to travel 400 meters.

Derive the boat's kinematics by obtaining position as a function of time and velocity as a function of position, using two sections to reach 400 m in 16.9 s.

Express position as a function of time for the boat and derive velocity and acceleration using ds/dt = v and dv/dt = a, with exponentials and ln.

Solve a 2D helicopter kinematics problem by deriving position, velocity, and acceleration from x and y functions in a fixed frame. Compute magnitudes at 10 seconds using Pythagoras.

Explore three-dimensional motion by using parametric equations as the position vector in time, controlling x, y, and z with separate functions to design spirals and variable ascent.

Investigate projectile motion by determining the initial bullet speed va for a shot at 30 degrees, assuming gravity 9.81 m/s^2 and no air resistance, using the given dimensions.

Solve a two-dimensional projectile problem by using constant-acceleration kinematic equations to relate the initial velocity components, angle 30 degrees, and gravitational acceleration to a 20-meter horizontal range.

Solving the falling box problem on a conveyor, lecture asks minimum and maximum distance to place 1 metre wide loading car moving at 2 m/s so that packages land inside.

Explain the falling box problem under gravity, using constant-acceleration kinematics for horizontal and vertical motion, derive initial vertical velocity from -30°, and compute final vertical velocity and travel time.

Apply horizontal kinematics with s_x = s_x0 + v0x t and v0x = v0 cosine 30 degrees to determine the box's reach R and the minimum and maximum R for catching it.

Examine a moving normal-tangential frame, using ut and un, and derive instantaneous velocity as ds/dt times ut, the speed along the curved trajectory.

Develop intuition for acceleration in rotating frames by decomposing into tangential and centripetal components using u t and u n, and theta dot, with a train and boat exercise.

Decompose acceleration into tangential and normal components to solve the train and boat problems, compute a_t and a_n, and use v^2/a_n for rho.

Derive particle motion in polar coordinates with varying radius, using radial and transverse components ur and u_theta, and show how changing angle alpha makes unit vectors rotate, affecting velocity.

Learn to decompose velocity and acceleration in polar coordinates, identifying v_r, v_theta, a_r, and a_theta, and apply to a car-camera tracking problem to find theta dot.

Compute theta dot for car in polar coords with r = 100 cos(2 theta); Use v^2 = r dot^2 + (r theta dot)^2 to obtain theta dot ≈ 0.378 rad/s.

Compute velocity and acceleration magnitudes for a box on a helical path in cylindrical coordinates; at theta = 2π, v ≈ 4.16 m/s and a ≈ 33.1 m/s^2.

Analyze pulley kinematics by modeling rope length with datum lines and differentiating to link end motion to block motion, yielding v_B = -1/2 v_A and a_A = -2 a_B.

Use a fixed datum line to derive velocity relations for three rope sections, form a matrix equation, and solve for vb, vd, and ve given va = 14 m/s upward.

Analyze relative velocities and accelerations in pulley systems by deriving times for a 3-meter rise and the velocity of A relative to B using reference-frame principles.

Solve a pulley problem using rope-length constraints to relate relative velocities and accelerations, yielding VB = -1/2 VC and VA = -1/2 VD with B up at 2 m/s.

Explain relative velocities and accelerations by solving block A rising three meters (t ≈ 1.22 s; VA ≈ -4.9 m/s) and measuring A relative to B via VA - VB, with car examples.

Determine plane A’s velocity relative to plane B by combining the 200 km/h airspeed with plane B’s 175 km/h relative to the ship at a 15-degree angle, with no wind.

explore relative motion on highways by analyzing cars a, b, and c with given speeds and accelerations, and determine the velocity and acceleration of car b relative to car c.

Compute v_B/C = 7.5 i + 17 j; include centripetal acceleration on curved road, yielding a_B ≈ 0.95 i + 2.86 j and a_B/C ≈ 0.95 i − 0.14 j.

Watch this quick follow up that shows how python animations illuminate dynamics concepts in engineering mechanics, with downloadable animation files at section eight and videos on installing python libraries.

Analyze a sliding box on a ramp to find the cord tension keeping cart A stationary while box B slides, using free body and kinetic diagrams and motion equations.

solve the sliding cart and box problem to determine rope cd tension on a frictionless ramp. use free body diagrams and decompose gravity into parallel and perpendicular components.

Derive x-y dynamics for cart a and a box on a ramp, relate cable tension to the normal force, and express it as (m box g)/2 sin 2 theta.

Solve for crate velocity at 2 seconds under a pulley system that doubles the motor force; the crate starts moving when the motor force balances gravity at about 1.77 seconds.

Compute the velocity of a cargo box lifted by a variable motor force through a pulley, using free body and kinetic diagrams to derive the acceleration after 1.7 seconds.

Use simplifying assumptions for pulleys and cables to simplify analysis: assume massless cables, negligible stretch, and constant tension, enabling clear free body and kinetic diagrams and straightforward equations of motion.

Draw the free body and kinetic diagrams for a pulley, assume no slipping under static friction, and derive motion equations with mass moment of inertia I and angular acceleration alpha.

Compare equal-mass objects to reveal how mass moment of inertia distribution affects angular acceleration, then assume a massless pulley and cable to derive T1 = T2 and C = 2T.

examine a tractor lifting a 150 kg load with a rope and pulley while moving at 4 m/s, and determine the rope tension when the length is 5 m.

Apply a free body diagram to compute rope tension for a 150 kg box, relate rope length with Pythagoras, and use chain and product rules under constant truck speed.

Derive AB and VB from rope-length equations with s = 5 m and v = 4 m/s; AB ≈ 1.05 m/s^2, VB ≈ 1.54 m/s, tension ≈ 1629 N.

tackle a two-block pulley system with equal masses and mu_k, where a horizontal force P moves the bottom block to determine block A's acceleration in case A and case B.

Solve the two-block and pulley problem by drawing free body diagrams, applying Newton's third law, and deriving block A's horizontal acceleration under kinetic friction and applied force P.

Analyze the acceleration of block a and block b in case b of a two-block pulley system with kinetic friction, using a fixed datum, free-body diagrams, and tension relations.

Define the leftward positive direction and derive the equations of motion for blocks A and B in a two-block pulley system. Include tension, kinetic friction mu k, and normal forces.

Expose a common trap in the two-block pulley with friction, showing accelerations are opposite and already accounted for in the kinetic diagrams, then derive the correct tension and acceleration.

Derive the velocity function for a parachutist who opens the parachute at rest, under drag proportional to v squared. Identify terminal velocity as time goes to infinity.

Solve the skydiving terminal velocity problem by deriving velocity as a function of time under quadratic drag, then determine the terminal velocity as time approaches infinity.

Explore valley and hill problem by finding the maximum speed to maintain contact over the hill top and the normal force at the bottom, using tangential, normal, and B coordinates.

Investigate a frictionless skier on a parabolic slope, derive the instantaneous center of rotation along a curve, and obtain the normal force at point B using the radius from ds/dx.

Compute the skier problem distance to the instantaneous center of rotation by applying chain and product rules to tan theta, deriving d^2y/dx^2, and forming R as a positive distance.

Determine the normal force at point B by using a free body diagram, accounting for gravity and centripetal acceleration via the radius and velocity.

Derive the skier's velocity and velocity squared by linking tan theta to x/10, expressing tangential acceleration as g x/(x^2+100), and applying v dv = a_t ds along the path.

Derive the skier's velocity at B by integrating V dV equals - (g/10) x dx to get VB = sqrt(10 g); then compute centripetal acceleration and the normal force.

Solve for lift and radius in a horizontal airplane turn: balance lift with gravity at 15°, compute centripetal acceleration, and obtain a turn radius of about 3594 meters.

Analyze a 2000 kg car on a curved road using polar coordinates to determine the resultant tire friction as the camera at A rotates with angular velocity and angular acceleration.

Analyze net forces in radial, theta, and z directions for a 250 kg roller coaster on a spiral track at time 2 s, using theta and z functions.

Solve a double embedded spring problem in dynamics using work and energy, with a frictionless block of 0.5 m/s stopping 0.3 m from the wall, to find spring b stiffness.

Explore how work arises from force and displacement via the dot product, a scalar, and how moments produce work through angular displacement, including positive and negative work with torsion springs.

Decompose forces on a particle into tangential components and apply the work-energy principle to relate total work to changes in kinetic energy.

Explain work due to friction by analyzing a sliding box with kinetic friction opposing the push, showing how heat loss means the ideal work-energy equation is incomplete.

Apply the work-energy principle to a frictionless box that stops 0.3 m from the wall, using two springs to determine KB from 225 J initial energy.

Analyze a 1.5 kg mass with velocity 4 m/s colliding with a nonlinear spring on a frictionless surface; f = k s^2 with k = 900 N/m^2, 0.2 m compression.

Use the work-energy principle to determine the box's final velocity as the nonlinear spring compresses 0.2 m, by equating initial and final kinetic energy with the spring's negative work.

Examine a dual-spring setup with a 20 kg mass on a smooth rod, where a 100 N force at 60 degrees drives motion to 0.3 m and yields the speed.

Apply a free-body diagram and the work-energy principle to analyze a mass with gravity, springs, and an applied force, finding the velocity when center point reaches the apostrophe, 2.36 m/s.

analyze projectile motion using work and energy: a 30 kg ball released from a swinging rope at theta=30 degrees computes its impact velocity and total distance to the ground.

Use the work-energy principle to analyze the projectile motion of a pendulum-like ball released from rest; gravity does work, yielding about 17.7 m/s at D after a 16 m drop.

Use work and energy to find the velocity at C from SBC; decompose into components, solve for time to D, and calculate R as 32.98 meters.

designing a roller coaster to achieve zero G at point C and four G at point B by selecting heights h, c, and h_A, starting from rest at A.

solves roller coaster problem to achieve zero g at c and four g at b, using free-body and work-energy to find h_A of 22.5 m and h_C of 12.5 m.

Explore a catapult mechanism that propels a 12 kg slider along a smooth track, driven by a piston delivering 30 kN for 0.3 m, starting from rest.

Apply the work and energy principle to the catapult, treating the slider as the only mass, 12 kilograms, and the 0.3 m movement to determine the final velocity.

Present a two-mass pulley system, show gravity does work on both masses, derive final kinetic energy in terms of v_a and v_b, and establish v_b = -2 v_a.

Analyze a double mass pulley using energy and constraint relations to reveal opposite accelerations with a_b = -2 a_a and a_a ≈ 1.98 m/s down, plus the cable tension.

Determine the compression of each spring in a two-spring series as a 10,000 kg train car moving at 5 m/s comes to rest, with rolling resistance 1/100 of weight.

Apply the work-energy principle to a train car colliding with two springs in series, account for initial kinetic energy and rolling resistance, and solve for S1 and S2.

Explore gravity's potential energy and the work it can store or release, and explain elastic potential energy in springs, with E = 1/2 k x^2, always positive.

Describes forming the total potential energy Vg + Ve, analyzes gravity and spring work, and states energy conservation for conservative forces.

Revisits the skier problem to compute velocity at point B and the normal force using conservation of energy, showing gravity converts potential energy into kinetic energy when friction is absent.

Apply energy conservation to a bungee jumper, incorporating gravity and elastic cord work to compute the cord stretch and stop position.

Analyzes the minimum height a roller coaster must start from to complete a loop without leaving the track and the velocity at point C at that height, neglecting friction.

Compute the minimum height for a roller coaster to complete a loop by balancing gravity and centripetal acceleration at the top, using energy conservation to find velocities.

Compute the velocity magnitude at point B for a 60 kg satellite in an elliptical orbit around Earth, using gravity, Earth's mass, and the universal gravitational constant.

Apply energy conservation to a space satellite, deriving velocity at point B from kinetic and gravitational potential energies, using a non-constant gravity work integral for potential energy.

Explore potential and kinetic energies for space satellites under a constant gravity model, using datum references and energy conservation to compute orbital velocity at point b.

Compare two cases with a 10 kg box starting from rest on a smooth surface, pulled by a constant force yielding 50 joules of work and the same final velocity.

Explain power as the rate of work, defined by power equals force times velocity, and show that the same work can occur faster in different cases, yielding different watts.

Understand power input and output in motors, how internal friction creates heat losses, and that 1000 W input at 0.9 efficiency yields 900 W output.

Compute the motor’s power output at t = 5 s for a 150 kg crate with static and kinetic friction 0.3 and 0.2, driven by a time-dependent leftward cable force.

Analyze a crate in a pulley system to determine the motor’s output power at five seconds, using a free body diagram with static and kinetic friction.

Calculate the rocket sled's power output over time for a 4000 kg mass on a horizontal track with friction 0.2 and thrust 150 kN, neglecting fuel loss and drag.

Solve for the sled's power output as a function of time using thrust and kinetic friction to get acceleration and velocity; obtain P(t) ≈ 5.33 t MW.

Calculate motor power input for a 50 kg crate moving up a 30-degree ramp, at v = 4 m/s when s = 8 m, neglect friction with 0.74 efficiency.

Determine the motor input power for a crate moving up a ramp by analyzing tension, acceleration, and velocity, and applying efficiency to yield about 1.6 kilowatts.

Determine the motor power to maintain a velocity of four meters per second for a cargo elevator with a 400 kg total mass and a 60 kg counterweight, given 0.6 efficiency.

Determine motor power for a pulley elevator with a counterweight to lift at four meters per second with no acceleration, deriving tension, cable velocity, and input power at 0.6 efficiency.

Apply the principle of linear impulse and momentum to an aircraft carrier thrust problem, relating initial and final momentum via impulse from time-varying thrust, using vector components.

Apply the linear impulse and momentum principle to an aircraft carrier problem: compute initial momentum L1, determine impulse from force over time, and use L1 plus impulse to find v2.

Calculate the airplane's horizontal velocity after 15 seconds using impulse from the two-engine thrust on a 250,000-kg jet, starting at 100 m/s and neglecting air resistance and fuel loss.

Apply the impulse momentum principle to find the airplane velocity at 15 seconds by integrating the total thrust from 0 to 15 seconds, starting from 100 m/s, yielding 142 m/s.

Engage with a crate and counterweight pulley system, a 200 kg crate and a 75 kg cylinder, to determine velocities at three seconds given the crate's initial velocity is zero.

Analyze a crate and counterweight pulley system to find the velocities of the cylinder and crate at 3 seconds using a free-body diagram, with the cylinder four times faster.

Apply the linear impulse–momentum principle to a crate and counterweight pulley, deriving cylinder and box velocities and solving for the tension and the opposite, 4:1 velocity relationship.

Explore a crate and counterweight pulley system using two free-body diagrams to derive tension, show a equals minus four times acceleration B, and verify with linear impulse and momentum method.

Calculate the time for a 50 kg crate on a 30-degree incline, with static and kinetic friction 0.3 and 0.2, to reach 2 m/s up ramp under a parallel force.

Explore the conservation of linear momentum in car collisions by using a control volume to show that internal forces cancel and horizontal external impulses vanish, yielding momentum conservation.

Explore how momentum remains conserved in collisions of two particles, even as energy is lost to friction, heat, and plastic deformation, while total work must be zero for energy conservation.

Explore impulsive versus non impulsive forces in a control volume, using impulse-momentum reasoning to decide when external weights or resistive forces alter momentum, with real-world examples.

The lecture uses momentum conservation to solve a three-car collision: first find AB's common velocity, then include C to obtain the final velocity, which is -55/7 km/h, implying leftward motion.

Analyze the canoe problem by tracking the boy’s jump from canoe a to canoe c at five meters per second relative velocity and establish the final velocities of both canoes.

Explore a collision of two freight cars A and B with a spring between them, and determine the maximum spring compression and the energy loss when they stick together.

The lecture analyzes hammer-pile impact, momentum conservation, coefficient of restitution, energy loss, and frictional descent into sand, and contrasts sticking versus bouncing collisions with deformation and restitution impulses.

Derives the linear impulse and momentum relations for deformation and restitution in a two-body impact, focusing on particle A and the coefficient of restitution E.

compute the coefficient of restitution from deformation and restitution impulses for two colliding particles, express e using initial and final velocities, and note momentum conservation with energy loss.

Use energy concepts to compute the hammer’s pre-impact velocity, then apply momentum with a 0.1 coefficient of restitution to find pile and hammer velocities, revealing a near-plastic impact.

Explore how the restitution coefficient E governs energy loss in a hammer-pile impact, noting momentum conservation for all E and energy conservation only at E = 1.

Apply the work-energy principle to determine the pile travel distance after impact, using initial speed 0.94 m/s, resistive force 18 kN, and gravity, yielding about 0.0348 m.

Analyze a two-ball pool collision: ball a hits ball b at rest (m=0.4 kg, e=0.8), then b rebounds from a wall (e=0.6) to yield b's velocity and theta.

Apply momentum conservation and restitution to solve a two-collision problem, yielding A's 0.5 m/s and B's 4.5 m/s at 30 degrees after impact.

Analyze the ball-wall impact by decomposing velocity into horizontal and vertical components, applying impulse within the ball’s volume, with restitution 0.6 and angle rising from 30 degrees to 43.9 degrees.

Explore angular impulse and momentum for a two-mass rod system on the xy plane, using r cross p and the right-hand rule to find angular momentum about the z axis.

Apply the principle of angular impulse and momentum by differentiating angular momentum and linking net moment to its change. Explore conservation and internal moments in coupled bodies.

Apply angular impulse and momentum principle to a two-mass rod system from rest, noting zero angular momentum and compute velocities at t = 2 s, yielding v2 ≈ 6.15 m/s.

Analyze a disk rotating about Point A on a plane attached to a spring-like cord; as the cord shortens from 1.5 m to 1.2 m, determine speed and radial rate.

Conserve angular momentum about point A on the z axis as the radius changes from 1.5 to 1.2 m; initial and final angular momentum yield v perpendicular = 6.25 m/s.

Use conservation of energy to find the resultant speed and parallel velocity along the elastic cord, illustrating angular momentum about the z axis is conserved while linear momentum is not.

Analyze steady, incompressible mass flow through a control volume, where volumetric flow equals velocity times area and remains constant from section A to B.

Apply the impulse–momentum approach to steady fluid flow in a control volume, deriving net external force as mass flow rate times the velocity difference, with angular momentum considerations.

Compute horizontal and vertical reactions and the fixed support moment for a hydrant flow of 0.75 cubic meters per second split between 75 millimeter outlets under 300 kilopascals gauge pressure.

Define a control volume around hydrant, convert gouge pressure to a 5301 N force, Qa = Qb = 0.375 m3/s with 84.88 m/s at A/B and 42.44 m/s at C.

Apply momentum and angular impulse concepts to a hydrant with one inlet and two outlets, deriving the net force and moment equations and a special case where inlet equals outlets.

Apply the equations of motion to compute horizontal and vertical reaction forces and moments at C in the hydrant problem, using mass flow rates and velocity components.

Compute hovercraft thrust using the linear impulse–momentum principle for steady, incompressible air. Analyze a 20 m3/s intake with 200 m/s horizontal and 800 m/s vertical exits, neglecting intake velocity.

Derive thrust by analyzing fluid flow relative to the moving propeller, using relative velocities and mass flow rate to relate force, acceleration, and exit and inlet speeds to the propeller.

Apply control-volume momentum balance to a hovercraft with 20 m³/s intake and 800 m/s exit velocity to determine mass flow, vertical equilibrium, and thrust generation.

Compute hovercraft thrust from steady air flow by applying mass flow rate and velocity difference between inlet and exit, assuming incompressible air and no horizontal external forces.

Compute reactions at D for a 0.3 m pipe bend carrying 1.35 m^3/s at 45 degrees, with 2500 N weight and 120/96 kN/m^2 gouge pressures at A and B.

Derive horizontal and vertical reaction forces and moments for a 45-degree water pipe using mass flow rate, density, specific weight, cross-sectional area, and steady-flow equations.

Compute the reaction moment by summing external moments about point D, using horizontal and vertical force components. Show cancellations of equal cosine and sine terms and solve for MD.

Explore variable mass propulsion using linear impulse and momentum for a rocket. Derive the control volume momentum balance with the velocity of the exiting mass and the device's motion.

Explore propulsion with variable mass through a rocket example, showing thrust from exhaust via Newton's third law, and external forces like weight and drag with relative velocity.

Explore a device that collects mass while moving, derive the variable-mass momentum equation from a control volume, and relate impulse to mass flow and relative velocity with the collected substance.

Analyze a two-stage rocket dynamics exercise analyzing stage b ignition and separation with fuel burn at 50 kg/s and exhaust velocity 2500 m/s to find maximum velocity of stage a.

Compute the maximum velocity of stage B in a two-stage rocket during a 10-second burn, yielding vb max = 165.3 m/s and stage B separation.

Compute the tractive force at the front wheels of a sand truck dumping sand at 900 kg/s, with 0.1 m/s^2 forward acceleration and empty mass 30,000 kg.

Compute the drag force on a jet plane in steady flight using air intake rate, fuel burn, exhaust velocity, and incompressible air with density 1.22 kg/m^3.

Compute the jet plane thrust and drag by analyzing mass flow, fuel addition, and relative air velocities, showing drag equals thrust for a constant horizontal speed.

Analyze the acceleration of a hovering firefighting helicopter immediately after releasing water from a 500 kg bucket at 50 kg/s with 10 m/s relative velocity, starting from rest.

Compute the helicopter's instantaneous acceleration after water release using lift minus weight on a 10,500 kg system, with water exiting at 50 kg/s and 10 m/s relative velocity.

Outline two impulsive delta-v maneuvers at A and B to switch from orbit D to a circular SE, then C to S, using apogee geometry and eccentricity 0.58.

Demonstrates that area swept by radius is constant, linking Kepler's second law to deriving r as a function of theta via the chain rule and h = r^2 theta_dot.

Derive the radial equation of motion in orbital mechanics, redefine with psi = 1/r, and frame gravity as a central force, yielding a differential equation whose solution is a function.

Explore solving a linear second order non-homogeneous differential equation by transforming to a homogeneous form, applying Euler's formula to express the solution in sines and cosines, and verifying via substitution.

Explore conic sections in polar coordinates, define eccentricity as the constant ratio r over directrix distance, and classify orbits: circle, ellipse, parabola, hyperbola, and the limiting line via e.

Learn how orbital mechanics link eccentricity to trajectory, compute radius from a focus, and compare circular and elliptical orbits with perigee, apogee, and velocity variation.

Explore how elliptical, parabolic, and hyperbolic trajectories depend on velocity at perigee and eccentricity, deriving escape velocity and circular orbit conditions.

Derives formulas for escape velocity, circular velocity, and orbital apogee from perigee velocity and eccentricity, explaining parabolic and crash trajectories and how eccentricity values shape orbit.

Express the ellipse's major axis A and minor axis B from perigee and apogee. Use area, angular momentum h, and the orbital period to plan a transfer to circular orbit.

Compute the velocity changes required to transfer a rocket from elliptical orbit d to orbit c by relating the perigee and apogee, eccentricity, and angular momentum to determine vad.

Compute the perigee velocity for orbit C from apogee data, then determine the required delta-v for the transfer from orbit D to C.

Conclude the dynamics course with python animations that clarify concepts, show how to run animation files, and provide installation videos for Ubuntu, Windows 10, and Mac OS, plus download files.

Install python and essential libraries on Windows 11, configure path, verify with Hello World, and install numpy, matplotlib, scipy, cvxopt, sympy, and control for MPC tasks and LQR.

Install python 3.8 and numpy on Ubuntu 20.04, verify versions, install pip and matplotlib, and run python scripts from the terminal for dynamics simulations.

install python 3.8.7 on macOS using the macOS 64-bit installer, verify with a terminal hello world test, install pip and numpy, then install matlab.