Question

The figure below shows the straight path of a particle across an xy coordinate system as it is accelerated from rest during time interval Δt1. The acceleration is constant. The xy coordinates for point A are (3.00 m, 4.00 m) and those for point B are (10.5 m, 20.0 m). (a) What is the ratio ay/ax of the acceleration components? (b) What are the coordinates of the particle if the motion is continued for another 3 intervals of Δt1?

Answer 1

consider the motion along the x-direction

X_o = initial position = 3 m

X_f = final position = 10.5 m

V_ox = initial velocity = 0

a_x = acceleration

using the equation

X_f = X_o + V_ox t + (0.5) a_x t²

10.5 = 3 + 0 t + (0.5) a_x t²

a_x = 15/t²

consider the motion along the y-direction

Y_o = initial position = 4 m

Y_f = final position = 20 m

V_oy = initial velocity = 0

a_y = acceleration

using the equation

Y_f = Y_o + V_oy t + (0.5) a_y t²

20 = 4 + 0 t + (0.5) a_y t²

a_y = 32/t²

ratio = a_y/a_x = (32/t²)/(15/t²) = 2.13

b)

for x-coordinate

X_f = X_o + V_ox (4t) + (0.5) a_x (4t)²

X_f = 3 + 0 (4t) + (0.5) (15/t²) (16 t²)

X_f = 123 m

for y-coordinate

Y_f = Y_o + V_oy (4t) + (0.5) a_y (4t)²

Y_f = 4 + 0 (4t) + (0.5) (32/t²) (16 t²)

Y_f = 260 m