The position of a particle moving along the x axis is given by x= 12t2 -2t ,where x is in meters and t in seconds . determine the velocity and the acceleration of the particle at t=3.0s
well, you have
x = 12t^2 - 2t
v = x' = 24t - 2
a = v' = 24
so plug in your numbers
X = 12T^2 - 2T
X = 12*3^2 - 2*3 = 102 m.
V = x/T = 108m/3s = 36 m/s.
V = a*T
36 = a*3
a =
To find the velocity and acceleration of the particle at t = 3.0s, we need to differentiate the position equation with respect to time.
Given:
Position equation, x = 12t^2 - 2t
Step 1: Find the velocity
Velocity is the derivative of position with respect to time.
v = dx/dt
Differentiating with respect to t:
v = d/dt(12t^2 - 2t)
v = 24t - 2
Step 2: Find the acceleration
Acceleration is the derivative of velocity with respect to time.
a = dv/dt
Differentiating with respect to t:
a = d/dt(24t - 2)
a = 24
Therefore, at t = 3.0s:
Velocity (v) = 24t - 2
Acceleration (a) = 24
To determine the velocity and acceleration of the particle at a specific time, you'll need to take the derivative of the position equation with respect to time.
Given:
x = 12t^2 - 2t
1. Velocity:
To find the velocity, you need to take the derivative of x with respect to t. The derivative of x with respect to t represents the rate of change of position with respect to time. So, differentiate x with respect to t:
v = dx/dt
First, differentiate each term separately:
d/dt (12t^2) = 24t
d/dt (-2t) = -2
So, the velocity equation is:
v = 24t - 2
2. Acceleration:
Similarly, to find the acceleration, you need to take the derivative of the velocity equation with respect to t. Again, the derivative of v with respect to t represents the rate of change of velocity with respect to time. Differentiate v with respect to t:
a = dv/dt
Differentiate each term separately:
d/dt (24t) = 24
d/dt (-2) = 0 (since any constant term has a derivative of zero)
So, the acceleration equation is:
a = 24
Now, to find the velocity and acceleration at t = 3.0s, substitute the value of t into the velocity and acceleration equations:
Velocity (v) at t = 3.0s:
v = 24t - 2
v = 24(3.0) - 2
v = 72.0 - 2
v = 70.0 m/s
Acceleration (a) at t = 3.0s:
a = 24
Therefore, at t = 3.0s, the velocity of the particle is 70.0 m/s and the acceleration is 24 m/s^2.