Motion(calculations) Flashcards
A toy car moves along a straight path with position given by x(t) = 3t + 0.5t^2, where x is in meters and t is in seconds. What is its average velocity from t = 0 s to t = 4 s?
Average velocity = (x(4) - x(0)) / (4 - 0). x(0) = 0, x(4) = 3(4) + 0.5(4^2) = 12 + 8 = 20 m. So, 20 / 4 = 5 m/s.
A ball’s position is x(t) = 4t - 0.2t^3 (meters, seconds). What is its instantaneous velocity at t = 2 s?
Velocity = derivative of x(t) = 4 - 0.6t^2. At t = 2, v = 4 - 0.6(2^2) = 4 - 0.6(4) = 4 - 2.4 = 1.6 m/s.
A rocket’s height is y(t) = 50 - 2t + 0.8t^2 (meters, seconds). What is its velocity at t = 3 s?
Velocity = derivative of y(t) = -2 + 1.6t. At t = 3, v = -2 + 1.6(3) = -2 + 4.8 = 2.8 m/s.
A cyclist moves with x(t) = 1.5t^2 - 0.1t^3 (meters, seconds). How long after starting does their velocity become zero?
Velocity = derivative of x(t) = 3t - 0.3t^2. Set v = 0: 3t - 0.3t^2 = 0. Factor: t(3 - 0.3t) = 0. Solutions: t = 0 or 3 - 0.3t = 0, so t = 10 s.
A drone’s position is x(t) = 2t + 0.4t^2 - 0.05t^3 (meters, seconds). What is its acceleration at t = 5 s?
Velocity = 2 + 0.8t - 0.15t^2. Acceleration = derivative of v = 0.8 - 0.3t. At t = 5, a = 0.8 - 0.3(5) = 0.8 - 1.5 = -0.7 m/s^2.
A train accelerates with velocity v(t) = 3 + 0.6t (m/s, seconds). What is its position at t = 10 s if it starts at x = 0?
Position = integral of v(t) = 3t + 0.3t^2 + C. At t = 0, x = 0, so C = 0. x(10) = 3(10) + 0.3(10^2) = 30 + 30 = 60 m.
A boat moves with v(t) = 5 - 0.2t^2 (m/s, seconds). What is its average acceleration from t = 0 s to t = 5 s?
a_avg = (v(5) - v(0)) / (5 - 0). v(0) = 5, v(5) = 5 - 0.2(5^2) = 5 - 5 = 0. a_avg = (0 - 5) / 5 = -1 m/s^2.
A particle’s position is x(t) = 0.7t^3 (meters, seconds). What is its velocity at t = 3 s?
Velocity = derivative of x(t) = 2.1t^2. At t = 3, v = 2.1(3^2) = 2.1(9) = 18.9 m/s.
A sled’s position is x(t) = 10 + 2t - 0.3t^2 (meters, seconds). What is its maximum distance from the start?
Velocity = 2 - 0.6t. Set v = 0: 2 - 0.6t = 0, t = 3.33 s. x(3.33) = 10 + 2(3.33) - 0.3(3.33^2) = 10 + 6.66 - 3.33 = 13.33 m.
A car’s velocity is v(t) = 4t - 0.1t^2 (m/s, seconds). What is its acceleration at t = 2 s?
Acceleration = derivative of v(t) = 4 - 0.2t. At t = 2, a = 4 - 0.2(2) = 4 - 0.4 = 3.6 m/s^2.
A ball is thrown upward with initial velocity 15 m/s from a height of 0 m. How high does it go? (Use g = 9.8 m/s^2)
At max height, v = 0. v = 15 - 9.8t, so t = 15 / 9.8 = 1.53 s. y = 15t - 4.9t^2, y(1.53) = 15(1.53) - 4.9(1.53^2) = 22.95 - 11.47 = 11.48 m.
A rock is dropped from a 20 m cliff. How long until it hits the ground? (g = 9.8 m/s^2)
A: y = -4.9t^2 = -20. t^2 = 20 / 4.9 = 4.08, t = 2.02 s.
A: y = -4.9t^2 = -20. t^2 = 20 / 4.9 = 4.08, t = 2.02 s.
A projectile is launched with horizontal velocity 10 m/s and vertical velocity 8 m/s. What is its horizontal range? (g = 9.8 m/s^2)
Time of flight: v_y = 8 - 9.8t, 0 = 8 - 9.8t, t = 0.816 s. Total time = 2 * 0.816 = 1.63 s. x = 10 * 1.63 = 16.3 m.
A stone is thrown at 12 m/s horizontally off a 5 m ledge. What is its speed when it hits the ground? (g = 9.8 m/s^2)
Time: y = -4.9t^2 = -5, t^2 = 5 / 4.9 = 1.02, t = 1.01 s. v_y = -9.8 * 1.01 = -9.9 m/s. Speed = sqrt(12^2 + (-9.9)^2) = sqrt(144 + 98) = 15.6 m/s.
A ball is kicked with initial velocity 20 m/s at 30 degrees. What is its maximum height? (g = 9.8 m/s^2)
v0y = 20 * sin(30) = 10 m/s. t = 10 / 9.8 = 1.02 s. y = 10 * 1.02 - 4.9(1.02^2) = 10.2 - 5.1 = 5.1 m.
A plane drops a package from 50 m with horizontal speed 30 m/s. How far does it land from the drop point? (g = 9.8 m/s^2)
Time: y = -4.9t^2 = -50, t^2 = 50 / 4.9 = 10.2, t = 3.19 s. x = 30 * 3.19 = 95.7 m.
A car in 2D moves with v = (6 - 0.1t^2) i + (0.4t) j (m/s, seconds). What is its acceleration at t = 4 s?
a_x = derivative of 6 - 0.1t^2 = -0.2t, a_x(4) = -0.8 m/s^2. a_y = derivative of 0.4t = 0.4 m/s^2. a = (-0.8 i + 0.4 j) m/s^2.
A robot moves with position r = (2t^2) i + (3t) j (meters, seconds). What is its speed at t = 3 s?
v_x = 4t, v_y = 3. At t = 3, v_x = 12 m/s, v_y = 3 m/s. Speed = sqrt(12^2 + 3^2) = sqrt(144 + 9) = 12.4 m/s.
A particle’s position is r = (5t - 0.05t^3) i + (2t^2) j (meters, seconds). What is its velocity at t = 2 s?
v_x = 5 - 0.15t^2, v_x(2) = 5 - 0.15(4) = 4.4 m/s. v_y = 4t, v_y(2) = 8 m/s. v = (4.4 i + 8 j) m/s.
A Ferris wheel has radius 10 m and spins at 0.5 rad/s. What is the tangential speed of a rider?
v = omega * R = 0.5 * 10 = 5 m/s.
A satellite orbits at radius 5000 m with tangential speed 20 m/s. What is its centripetal acceleration?
a = v^2 / R = 20^2 / 5000 = 400 / 5000 = 0.08 m/s^2.
A merry-go-round spins with period 4 s and radius 6 m. What is the tangential speed?
v = 2 * pi * R / T = 2 * 3.14 * 6 / 4 = 37.68 / 4 = 9.42 m/s.
A disk spins at 2 rad/s with a mass at radius 0.1 m. What force keeps a 0.2 kg mass from sliding?
a = omega^2 * R = 2^2 * 0.1 = 0.4 m/s^2. F = m * a = 0.2 * 0.4 = 0.08 N.
A car on a circular track (radius 50 m) has centripetal acceleration 8 m/s^2. What is its speed?
a = v^2 / R, v^2 = 8 * 50 = 400, v = 20 m/s.
Two cars move with x_A(t) = 2t + t^2 and x_B(t) = 3t^2 - 0.5t^3 (meters, seconds). When do they have the same velocity?
t = 2 or t = 2/3
A ball rolls with v(t) = 8 - 0.3t^2 (m/s, seconds). What is its position at t = 4 s if it starts at x = 0?
x(t) = integral of v = 8t - 0.1t^3. x(4) = 8(4) - 0.1(4^3) = 32 - 6.4 = 25.6 m.
A kite’s height is y(t) = 30 + 5t - 0.2t^2 (meters, seconds). When does it reach its maximum height?
v = 5 - 0.4t. Set v = 0: 5 - 0.4t = 0, t = 12.5 s.
A puck slides with v = (3t) i + (4 - 0.1t^2) j (m/s, seconds). What is its acceleration magnitude at t = 2 s?
a_x = 3 m/s^2, a_y = -0.2t, a_y(2) = -0.4 m/s^2. Magnitude = sqrt(3^2 + (-0.4)^2) = sqrt(9 + 0.16) = 3.02 m/s^2.
A pendulum bob moves in a circle (radius 1 m) with speed 2 m/s. What is its period?
v = 2 * pi * R / T, 2 = 2 * 3.14 * 1 / T, T = 6.28 / 2 = 3.14 s.
A rocket’s position is y(t) = 100 - 3t + 0.5t^2 (meters, seconds). What is its velocity just before hitting y = 0?
y = 0: 100 - 3t + 0.5t^2 = 0. t = (3 ± sqrt(9 + 200)) / 1 = 17.46 s (positive). v = -3 + t, v(17.46) = -3 + 17.46 = 14.46 m/s.
