The vector position of a particle varies in time according to the expression r with arrow = 6.60 i − 9.00t2 j where r with arrow is in meters and t is in seconds.
(a) Find an expression for the velocity of the particle as a function of time. (Use any variable or symbol stated above as necessary.)
v with arrow =
m/s
(b) Determine the acceleration of the particle as a function of time. (Use any variable or symbol stated above as necessary.)
a with arrow =
m/s2
(c) Calculate the particle's position and velocity at t = 5.00 s.
r with arrow =
m
v with arrow =
m/s
a) The expression for velocity can be found by taking the derivative of the position vector with respect to time:
v with arrow = d(r with arrow)/dt = 6.60 i - 18.00t j
So, the expression for velocity of the particle as a function of time is v with arrow = 6.60 i - 18.00t j m/s.
b) The expression for acceleration can be found by taking the derivative of the velocity vector with respect to time:
a with arrow = d(v with arrow)/dt = 0 i - 18.00 j
So, the expression for acceleration of the particle as a function of time is a with arrow = -18.00 j m/s^2.
c) To find the particle's position and velocity at t = 5.00 s, we substitute the value of t into the given expressions:
r with arrow = 6.60 i - 9.00(5.00)^2 j
= 6.60 i - 9.00(25) j
= 6.60 i - 225.00 j
v with arrow = 6.60 i - 18.00(5.00) j
= 6.60 i - 90.00 j
Therefore, at t = 5.00 s, the particle's position is r with arrow = 6.60 i - 225.00 j m, and its velocity is v with arrow = 6.60 i - 90.00 j m/s.
To find the velocity of the particle as a function of time, we need to differentiate the position vector, r with arrow, with respect to time. Let's start with the position vector:
r with arrow = 6.60i - 9.00t^2j
(a) The velocity vector, v with arrow, is given by the derivative of the position vector with respect to time:
v with arrow = d(r with arrow)/dt
Taking the derivative of each component separately, we have:
v with arrow = d(6.60i)/dt - d(9.00t^2j)/dt
v with arrow = 0i - (18.00t)j
So, the expression for the velocity of the particle as a function of time is:
v with arrow = - 18.00tj m/s
(b) To determine the acceleration of the particle as a function of time, we need to differentiate the velocity vector, v with arrow, with respect to time:
a with arrow = d(v with arrow)/dt
Taking the derivative of each component separately, we have:
a with arrow = d(0i)/dt - d(18.00tj)/dt
a with arrow = 0i - 18.00j
So, the expression for the acceleration of the particle as a function of time is:
a with arrow = - 18.00j m/s^2
(c) To calculate the particle's position and velocity at t = 5.00 s, we can substitute t = 5.00s into the expressions we derived earlier.
Substituting t = 5.00s into the position vector:
r with arrow = 6.60i - 9.00(5.00^2)j
r with arrow = 6.60i - 9.00(25.00)j
r with arrow = 6.60i - 225.00j
So, the particle's position at t = 5.00s is:
r with arrow = 6.60i - 225.00j meters
Substituting t = 5.00s into the velocity vector:
v with arrow = - 18.00(5.00)j
v with arrow = - 90.00j
So, the particle's velocity at t = 5.00s is:
v with arrow = - 90.00j m/s
To find the velocity of the particle, we differentiate the position vector with respect to time.
(a) Differentiating the position vector r with respect to time, we can find the expression for velocity v:
v with arrow = (d/dt) r with arrow
Given r with arrow = 6.60i - 9.00t^2j
Differentiating the x-component: (d/dt)(6.60i) = 0
Differentiating the y-component: (d/dt)(-9.00t^2j) = -9.00 * (d/dt)(t^2)j
Using the power rule for differentiation, (d/dt)(t^2) = 2t, so
v with arrow = 0i - 9.00 * 2tj = -18.00tj
Therefore, the expression for the velocity of the particle as a function of time is v with arrow = -18.00tj, where t is in seconds.
(b) To find the acceleration, we differentiate the velocity vector with respect to time.
a with arrow = (d/dt) v with arrow
Differentiating the y-component: (d/dt)(-18.00tj) = -18.00 * (d/dt)(t)j
Differentiating t with respect to t gives us 1, so
a with arrow = -18.00 * 1j = -18.00j
Therefore, the acceleration of the particle as a function of time is a with arrow = -18.00j, where j is the unit vector in the y-direction
(c) To calculate the particle's position and velocity at t = 5.00 seconds, we substitute t = 5.00 into the expressions we found earlier.
Substituting t = 5.00 into the position vector, we get:
r with arrow = 6.60i - 9.00(5.00^2)j = 6.60i - 9.00(25.00)j = 6.60i - 225.00j
So, at t = 5.00 seconds, the particle's position is r with arrow = 6.60i - 225.00j, where i and j are unit vectors in the x and y directions, respectively.
Substituting t = 5.00 into the velocity vector, we get:
v with arrow = -18.00(5.00)j = -90.00j
So, at t = 5.00 seconds, the particle's velocity is v with arrow = -90.00j, where j is the unit vector in the y-direction.
Do you know calculus?
a) take the first derivative of position to get velocity.
p(t)=6.60 i − 9.00t^2 j
v(t)=0i -18t j
b. velocity given above, so acceleration is dv/dt or
a(t)=-18 j
c. put in t=5 and calculate.