To find the time at which the particle first crosses the x-axis, we need to determine when the x-component of its position becomes zero.
Step 1: Find the x-coordinate equation.
The particle's position ⃗r as a function of time t can be expressed as:
⃗r(t) = ⃗r(0) + ⃗v(0)t + 0.5⃗at^2
Since ⃗a = −(2 m/s^2)(i + j), the equation becomes:
⃗r(t) = (4 m)j + (2 m/s)i * t + 0.5(-2 m/s^2)(i + j)t^2
Breaking it down into x and y components, we have:
x(t) = (2 m/s)t - (m/s^2)t^2
y(t) = 4 m + (2 m/s)t - (m/s^2)t^2
Step 2: Set x(t) = 0 and solve for t.
To find the time at which the particle crosses the x-axis, set x(t) = 0 and solve for t:
0 = (2 m/s)t - (m/s^2)t^2
Rearranging the equation, we get:
(m/s^2)t^2 - (2 m/s)t = 0
Factoring out t, we have:
t(t - 2 s) = 0
Setting each factor equal to zero, we get:
t = 0 s (This is the initial time when the particle is at position ⃗r(0))
t - 2 s = 0
Solving for t, we find:
t = 2 s
Therefore, the particle first crosses the x-axis at t = 2 seconds.
Now let's move on to part 2.
To find the time at which the particle is moving parallel to the y-axis (in the j direction), we need to determine when the x-component of its velocity becomes zero.
Step 1: Find the x-component of velocity.
The particle's velocity ⃗v as a function of time t can be expressed as:
⃗v(t) = ⃗v(0) + ⃗at
Since ⃗a = −(2 m/s^2)(i + j), the equation becomes:
⃗v(t) = (2 m/s)i + (-2 m/s^2)(i + j)t
Breaking it down into x and y components, we have:
v_x(t) = (2 m/s) - (2 m/s^2)t
v_y(t) = 0 m/s - (2 m/s^2)t
Step 2: Set v_x(t) = 0 and solve for t.
To find the time at which the x-component of velocity becomes zero, set v_x(t) = 0 and solve for t:
0 = (2 m/s) - (2 m/s^2)t
Rearranging the equation, we get:
(2 m/s^2)t = (2 m/s)
Simplifying, we find:
t = 1 s
Therefore, the particle is moving parallel to the y-axis at t = 1 second.