At t = 0, a particle moving in the xy plane with constant acceleration has a velocity of vi = (3.00 i - 2.00 j) m/s and is at the origin. At t = 2.00 s, the particle's velocity is v = (6.60 i + 4.00 j) m/s.
Find the acceleration of the particle at any time t. (Use t, i, and j as necessary.)
Subtract the two vector velocities from one another and divide the difference by 2.00 s to get the vector acceleration.
Since acceleration is said to be constant, it will be valid at all t.
To find the acceleration of the particle at any time t, we need to use the equation:
v = vi + at
Where:
v is the final velocity of the particle,
vi is the initial velocity of the particle,
a is the acceleration of the particle,
t is the time at which we need to find the acceleration.
Given that the initial velocity vi = (3.00 i - 2.00 j) m/s, and at t = 2.00 s, the velocity v = (6.60 i + 4.00 j) m/s, we can substitute these values into the equation to solve for a:
v = vi + at
(6.60 i + 4.00 j) m/s = (3.00 i - 2.00 j) m/s + a(2.00 s)
Separating the i and j components of the equation:
6.60 i + 4.00 j = 3.00 i - 2.00 j + 2a i
Equate the i components:
6.60 i = 3.00 i + 2a i
Combine like terms:
6.60 i - 3.00 i = 2a i
(6.60 - 3.00) i = 2a i
3.60 i = 2a i
Divide both sides by i:
3.60 = 2a
Now, we can determine the value of a by dividing both sides by 2:
a = 3.60 / 2
a = 1.80 m/s^2
Therefore, the acceleration of the particle at any time t is 1.80 m/s^2.
To find the acceleration of the particle at any time t, we can use the following formula:
v = vi + at
where:
v is the final velocity of the particle,
vi is the initial velocity of the particle,
a is the acceleration of the particle, and
t is the time.
Given that the initial velocity (vi) is (3.00 i - 2.00 j) m/s and the final velocity (v) at t = 2.00 s is (6.60 i + 4.00 j) m/s, we can substitute these values into the equation.
v = vi + at
(6.60 i + 4.00 j) m/s = (3.00 i - 2.00 j) m/s + a * 2.00 s
Now, let's separate the x and y components:
(6.60 i) m/s + (4.00 j) m/s = (3.00 i) m/s - (2.00 j) m/s + (ax i + ay j) * 2.00 s
Equating the x and y components separately, we get two equations:
6.60 = 3.00 + 2.00a_x
4.00 = -2.00 + 2.00a_y
Now, we can solve these two equations to find the x (a_x) and y (a_y) components of the acceleration.
From the first equation:
2.00a_x = 6.60 - 3.00
2.00a_x = 3.60
a_x = 1.80 m/s^2
From the second equation:
2.00a_y = 4.00 + 2.00
2.00a_y = 6.00
a_y = 3.00 m/s^2
Therefore, the acceleration of the particle at any time t is (1.80 i + 3.00 j) m/s^2.