A 1.57-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) = (7.00ti − 8.00j)N with t in seconds.
(a) At what time t will the particle's speed be 19.0 m/s?
s
(b) How far from the origin will the particle be when its velocity is 19.0 m/s?
m
(c) What is the particle's total displacement at this time? (Express your answer in vector form. Do not include units in your answer.)
The answer keyare: a)=2.51s, b)=19.9m, and c)=11.8i-16.I got my acceleration by dividing N with Kg=(4.46ti+5.10tj)m/s^2 but got stuck here.
(a) you want
(7.00t)^2 + 8.00^2 = 19^2
(b) now plug that t value and integrate twice of the F/m vector (acceleration) with v(0)=s(0)=0
(c) use the distance formula again for the new position
ehh Steve your method doesn't get me anywhere closer to the answer key. Cau you explain futher with steps?
Hmmm. I see I didn't read carefully.
F(t) = <7t,-8>
since F=ma,
a(t) = F/1.57 = <4.458t,-5.096>
so,
v(t) = ∫a(t) dt = <2.279t^2,-5.096t>
|v| = 19 when
(2.279t^2)^2 + (5.096t)^2 = 19^2
t = 2.49
Not sure how they got 2.51, but maybe some roundoff is involved.
Now integrate v to get s(t) and work from that.
To find the time at which the particle's speed is 19.0 m/s, we need to determine its acceleration first.
Given that the force applied on the particle is F(t) = (7.00ti - 8.00j) N, we can use Newton's second law: F = ma, where m is the mass of the particle and a is its acceleration.
Considering that the particle is initially at rest, we have:
F = ma
(7.00ti - 8.00j) = (1.57 kg) * a
To find the acceleration, we can equate the x-component and y-component of the force equation to the corresponding components of the mass times acceleration equation:
7.00ti = (1.57 kg) * ax ...(1)
-8.00j = (1.57 kg) * ay ...(2)
From equation (1), we can determine the x-component of the acceleration as:
ax = (7.00ti) / (1.57 kg)
Differentiating ax with respect to time t, we can find the x-component of the velocity:
vx = integral (ax) dt
= ∫ [(7.00ti) / (1.57 kg)] dt
= (7.00 / 1.57) * (t^2 / 2)
However, we need the particle's speed, which is the magnitude of velocity. So, we can find it as:
v = |v| = sqrt(vx^2 + vy^2)
v = sqrt[(7.00^2 / 1.57^2) * (t^2 / 2)^2]
Now, we can solve for t when v = 19.0 m/s:
19.0 = sqrt[(7.00^2 / 1.57^2) * (t^2 / 2)^2]
Squaring both sides and simplifying:
361.00 = [(7.00^2 / 1.57^2) * (t^2 / 2)^2]
To solve this equation, we can first isolate the term involving t:
(t^2 / 2)^2 = (361.00 * 1.57^2) / 7.00^2
Taking the square root of both sides:
t^2 / 2 = sqrt((361.00 * 1.57^2) / 7.00^2)
Finally, solve for t:
t = sqrt(2 * sqrt((361.00 * 1.57^2) / 7.00^2))
Evaluating this expression will give the value of t in seconds.