To determine the correct relationship between Δd1, v1, Δd2, and v2, let's analyze the problem step by step.
First, let's find the displacement (Δd1) and velocity (v1) after time t using the given acceleration components ax = a and ay = a.
We know that acceleration (a) is the derivative of velocity (v) with respect to time (t):
a = dv/dt
Integrating both sides with respect to time gives us:
v = at + C1
Since the object starts from rest, we have v=0 when t=0:
0 = a(0) + C1
C1 = 0
Therefore, the velocity v1 after time t is:
v1 = at
Next, let's calculate the displacement (Δd1) after time t using the velocity-time equation:
v = ds/dt
Integrating both sides with respect to time gives us:
s = vt + C2
Since the object starts from rest, we have s=0 when t=0:
0 = a(0)t + C2
C2 = 0
Therefore, the displacement Δd1 after time t is:
Δd1 = vt = at^2
Now, let's find the displacement (Δd2) and velocity (v2) after time t using the given acceleration components ax = 2a and ay = 2a.
Using the same approach as before, the velocity v2 after time t is:
v2 = (2a)t = 2at
And the displacement Δd2 after time t is:
Δd2 = (2at)(t) = 2at^2
Now we can compare the relationships between Δd1, v1, Δd2, and v2.
Δd2 = 4Δd1 and v2 = 2v1
Comparing the expressions we derived:
Δd2 = 2at^2 and Δd1 = at^2
v2 = 2at and v1 = at
Therefore, the correct relationship is:
Δd2 = 2Δd1 and v2 = 2v1
So the correct option is:
Δd2 = 2Δd1 and v2 = 2v1.