A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 24 m/s west,and the acceleration of the car has componets of 2.4 m/s^2 east and 1.8m/s^2 south. Find the total displacement of the object (from t=0 to 25)
To find the total displacement of the car, we need to consider the velocity and acceleration vectors at the initial instant and integrate the acceleration vector over time to find the change in velocity. Then, we can integrate the change in velocity vector over time to find the total displacement.
Given:
Velocity of the car (v) = 24 m/s west
Acceleration components (a_east, a_south) = (2.4 m/s^2 east, 1.8 m/s^2 south)
Step 1: Find the change in velocity over time
The acceleration vector can be split into its east-west (a_x) and north-south (a_y) components as follows:
a_x = 2.4 m/s^2 east
a_y = -1.8 m/s^2 (since south is in the negative y-direction)
The change in velocity over time can be calculated using the equations:
Δv_x = a_x * t
Δv_y = a_y * t
where t is the time interval.
Considering time interval t = 25 s, we find:
Δv_x = a_x * t = 2.4 m/s^2 * 25 s = 60 m/s east
Δv_y = a_y * t = -1.8 m/s^2 * 25 s = -45 m/s south
So, the change in velocity (Δv) is:
Δv = (60 m/s east, -45 m/s south)
Step 2: Find the total displacement
To find the total displacement, we need to integrate the change in velocity over time.
The displacement vector can be split into its left-right (x-direction) and up-down (y-direction) components as follows:
Δx = Δv_x * t
Δy = Δv_y * t
Considering t = 25 s, we find:
Δx = Δv_x * t = 60 m/s * 25 s = 1500 m east
Δy = Δv_y * t = (-45 m/s) * 25 s = -1125 m south
Therefore, the total displacement (Δr) is:
Δr = (1500 m east, -1125 m south)