can somebody shows me how to get to the answer key step by step. the answer key is 2.78i-10.2j A 1.49-kg particle initially at rest and at the origin of an x-y coordinate system is subjected to a time-dependent force of
F(t) = (4.00ti − 9.00j) N
with t in seconds.What is the particle's total displacement at 12.0 m/s?
F=ma, so
a(t) = 2.684ti-6.040j
v(t) = 1.342t^2 i - 6.040t j
s(t) = 0.447t^3 i - 3.020t^2 j
Do you mean when the speed is 12.0 m/s? If so, that occurs when
|v|=12.0
√((1.342t^2)^2 + (6.040t)^2) = 12.0
t = 1.839
s(1.839) = 0.447*1.839^3 i - 3.020*1.839^2 j = 2.78i-10.2j
what don't get it. ecan you explain more on the part √((1.342t^2)^2 + (6.040t)^2) = 12.0
t = 1.839 ?
To find the particle's total displacement, we need to integrate its velocity function over time.
Step 1: Start with the given force function:
F(t) = (4.00ti - 9.00j) N
Step 2: To find the particle's acceleration, we can use Newton's second law:
F(t) = m * a
where m is the mass of the particle, which is given as 1.49 kg. Rearrange the equation to solve for acceleration:
a = F(t) / m
Substituting the given force function and mass, we have:
a = (4.00ti - 9.00j) / 1.49
Step 3: Integrate the acceleration function to find the particle's velocity. Since the particle is initially at rest, we will need to determine the constant of integration.
∫a dt = ∫[(4.00ti - 9.00j) / 1.49] dt = v + C
Evaluating the integral, we obtain:
v = (2.00t^2)i - (9.00t)j + C
To determine the constant of integration (C), we can use the fact that the particle is initially at rest. Since v(0) = 0, substitute t = 0 into the velocity function and solve for C:
0 = (2.00(0)^2)i - (9.00(0))j + C
C = 0
So the velocity function becomes:
v = (2.00t^2)i - (9.00t)j
Step 4: Finally, integrate the velocity function to find the particle's displacement. Again, we will need to determine the constant of integration.
∫v dt = ∫[(2.00t^2)i - (9.00t)j] dt = r + C
Evaluating the integral, we obtain:
r = (0.67t^3)i - (4.50t^2)j + C
To determine the constant of integration (C), we can apply the initial condition that the particle is at the origin when t = 0. So r(0) = 0i - 0j = 0, leading to C = 0.
The displacement function becomes:
r = (0.67t^3)i - (4.50t^2)j
Step 5: Substitute the given velocity of 12.0 m/s for t in the displacement function:
r = (0.67(12.0)^3)i - (4.50(12.0)^2)j
r = (0.67(1728))i - (4.50(144))j
r = (1155.36)i - (648.00)j
Thus, the particle's total displacement is 1155.36i - 648.00j (units depend on the original unit of time).