An object moves with an initial velocity
vi = 3.30j m/s
and an acceleration
a = 2.60i m/s2.
Assume the object is initially at the origin.
(a) What is the position vector of the object as a function of time? (Express your answer in vector form. Use the following as necessary: t.)
r(t) =
m
(b) What is the velocity vector of the object as a function of time? (Express your answer in vector form. Use the following as necessary: t.)
v(t) =
m/s
(c) What is the position of the object at time
t = 3.50 s? (Express your answer in vector form.)
r(3.50 s) =
m
(d) What is the speed of the object at time
t = 3.50 s?
m/s
a(t) = 2.60 i
v(t) = ∫a dt + vi = 2.60t i + 3.30 j
r(t) = ∫v dt + 0 = 1.3t^2 i + 3.30t j
now finish it off, noting for (d) that the speed = |v|
Vi = 3.30 j
there is NO acceleration in the y direction so this is the y velocity forever.
therefore y = 3.30 t
A = 2.60 i
so
Vx = 2.60 t
x = 1.30 t^2/2 = 0.65 t^2
so
a = 2.60 i + 0 j
v = 2.60 t i + 3.30 j
position = 0.65 t^2 i + 3.30 t j
whoops 2.60 / 2 = 1.30 not 0.65
To find the position vector of the object as a function of time, we need to integrate the velocity function with respect to time.
(a) Position vector as a function of time:
To find the position vector, we integrate the velocity vector function.
Given:
Initial velocity, vi = 3.30j m/s
Acceleration, a = 2.60i m/s^2
Integrating acceleration with respect to time will give us the velocity vector:
v(t) = ∫ a dt = ∫ 2.60i dt = (2.60ti + C1)j
Where C1 is the constant of integration.
Integrating the velocity vector with respect to time will give us the position vector:
r(t) = ∫ v(t) dt = ∫ (2.60ti + C1)j dt = (1.30ti^2 + C1t + C2)j
Where C2 is the constant of integration.
So, the position vector as a function of time is:
r(t) = (1.30ti^2 + C1t + C2)j
(b) Velocity vector as a function of time:
v(t) = (2.60ti + C1)j
(c) Position of the object at time t = 3.50 s:
To find the position of the object at a specific time, substitute the given time value into the position vector function.
r(3.50 s) = (1.30(3.50)(3.50) + C1(3.50) + C2)j = (15.225 + 3.50C1 + C2)j
(d) Speed of the object at time t = 3.50 s:
To find the speed at a specific time, calculate the magnitude of the velocity vector at that time.
v(3.50 s) = (2.60(3.50)t + C1)j
The speed of an object is given by the magnitude of the velocity vector:
Speed = |v(3.50 s)| = √((2.60(3.50))^2 + C1^2) = √(30.45 + C1^2) m/s