Dynamics Problems - Problem 1.3

Problem 1.3: Between 2 stations, an underground train covers a distance of 3 km. Given are the starting acceleration a_a = 0.2 m/s^2, the braking deceleration a_d = -0.6 m/s^2, and the maximum speed v^* = 90 km/h. Determine the acceleration distance, the deceleration distance, the distance during uniform motion, and the travel time. Matlab Code: % Given data distance = 3 * 10^3; % distance between stations in meters aa = 0.2; % starting acceleration in m/s^2 ad = -0.6; % braking deceleration in m/s^2 v_star = 90 * (1000 / 3600); % maximum speed in m/s % To determine acceleration distance, deceleration distance, distance during uniform motion, and travel time, % we'll use equations of motion and kinematic equations % a) Determine acceleration distance % v = u + at, where u = 0 (initial velocity) % v_star = 0 + aa * t_acc t_acc = v_star / aa; distance_acc = 0.5 * aa * t_acc^2; % b) Determine deceleration distance % v^2 = u^2 + 2ad, where v = 0 (final velocity) % 0 = v_star^2 + 2 * ad * distance_dec distance_dec = v_star^2 / (-2 * ad); % c) Determine distance during uniform motion distance_uniform = distance - distance_acc - distance_dec; % d) Determine travel time % Total time is the sum of time for acceleration, time for uniform motion, and time for deceleration time_acc = t_acc; time_uniform = distance_uniform / v_star; % Final velocity during deceleration is 0, so time for deceleration can be calculated as: time_dec = abs(v_star / ad); total_time = time_acc + time_uniform + time_dec; % Display results fprintf('Acceleration distance: %.2f meters\n', distance_acc); fprintf('Deceleration distance: %.2f meters\n', distance_dec); fprintf('Distance during uniform motion: %.2f meters\n', distance_uniform); fprintf('Travel time: %.2f seconds\n', total_time); Problem 1.4: A car driver approaches a traffic light with a speed of v_0 = 50 km/h. At a distance of l = 100 m, the lights turn to 'Red'. The 'Red' and 'Yellow' phase takes t^* = 10 s. The driver wants to pass the traffic lights just when the lights turn back to 'Green'. a) Determine the necessary constant deceleration a_0 when the driver is braking along the entire distance. b) Determine the velocity v_1 of the car when arriving at the lights. c) Draw the diagrams a(t) , v(t) , and x(t) . Matlab Code: % Given data v0 = 50 * (1000 / 3600); % initial velocity in m/s l = 100; % distance to the traffic light in meters t_star = 10; % duration of 'Red' and 'Yellow' phases in seconds % a) Determine the necessary constant deceleration a0 when the driver is braking along the entire distance % The car should come to a stop just as it reaches the traffic light % Using the equation v^2 = u^2 + 2as % where v is final velocity (0), u is initial velocity, a is deceleration, and s is distance a0 = -(v0^2) / (2 * l); % b) Determine the velocity v1 of the car when arriving at the lights % During deceleration: v^2 = u^2 + 2as % where u is initial velocity, a is deceleration, s is distance, and v is final velocity v1 = sqrt(v0^2 + 2 * a0 * l); % c) Draw the diagrams a(t), v(t), and x(t) % Since the deceleration is constant, we can directly plot the equations % Time vector t = 0:0.1:t_star; % Acceleration over time a = a0 * ones(size(t)); % Velocity over time v = v0 + a0 * t; % Position over time x = v0 * t + 0.5 * a0 * t.^2; % Plotting subplot(3,1,1); plot(t, a); title('Acceleration over Time'); xlabel('Time (s)'); ylabel('Acceleration (m/s^2)'); subplot(3,1,2); plot(t, v); title('Velocity over Time'); xlabel('Time (s)'); ylabel('Velocity (m/s)'); subplot(3,1,3); plot(t, x); title('Position over Time'); xlabel('Time (s)'); ylabel('Position (m)'); % Display results fprintf('Necessary constant deceleration: %.2f m/s^2\n', a0); fprintf('Velocity of the car when arriving at the lights: %.2f m/s\n', v1); Problem 1.5: A vehicle moves according to the given speed-time diagram. Determine the occurring accelerations, the covered distance, and draw the diagrams x(t) , a(t) , v(x) , and a(x) . Matlab Code: % Given speed-time diagram time = [0, 5, 10, 15, 20]; % time points (s) speed = [0, 10, 10, 20, 20]; % corresponding speeds (m/s) % Calculate accelerations accelerations = diff(speed) ./ diff(time); % Calculate covered distance distances = cumtrapz(time, speed); % Interpolating for smoother plots time_interp = linspace(time(1), time(end), 100); speed_interp = interp1(time, speed, time_interp, 'linear', 'extrap'); distance_interp = interp1(time, distances, time_interp, 'linear', 'extrap'); % Calculate accelerations and velocities at each point accel_interp = [accelerations(1), accelerations, accelerations(end)]; velocity_interp = gradient(distance_interp, time_interp); % Plotting x(t), a(t), v(x), and a(x) subplot(2, 2, 1); plot(time_interp, distance_interp); title('Distance vs Time'); xlabel('Time (s)'); ylabel('Distance (m)'); subplot(2, 2, 2); plot(time_interp, accel_interp); title('Acceleration vs Time'); xlabel('Time (s)'); ylabel('Acceleration (m/s^2)'); subplot(2, 2, 3); plot(distance_interp, velocity_interp); title('Velocity vs Distance'); xlabel('Distance (m)'); ylabel('Velocity (m/s)'); subplot(2, 2, 4); plot(distance_interp, accel_interp); title('Acceleration vs Distance'); xlabel('Distance (m)'); ylabel('Acceleration (m/s^2)');