Plane Curvilinear Motion: x-y Coordinates - Example Problem 3.1-7

 

A girl operates a radio controlled model car on a smooth flat cement playground area. The girl stands at the origin of the x-y coordinate system and the surface of the cement lies in the x-y plane. The car’s position is given by r = xi + yj and the x- and y-components of the velocity are given by dx/dt = 2 + 3t² m/s and dy/dt = 6 + 0.5t³ m/s, where t is in seconds. At time t = 0, the car is located at x = 2 m and y = 6 m.

1. Determine, as a function of time, the position, velocity and acceleration of the car.
2. Plot the position of the car for the time period of t = 0 to t = 2 seconds.
3. Determine the car’s displacement Δr during the interval from t = 0 to t = 2 s.
4. Determine the velocity and acceleration of the car at t = 1 second and plot this on the position graph.
5. Determine the average velocity and acceleration of the car during the interval from t = 0 to t = 2 s.

 

 

Given:

• r = x i + y j
• dx/dt = 2 + 3t ² m/s
• dy/dt = 6 + 0.5t ³ m/s
• x(0) = 2 m
• y(0) = 6 m

 

Find:

1. r, v, a
2. plot position from t = 0 to 2 s
3. Δr between t = 0 and 2 s
4. v₁, a₁ @ t = 1 s and superimpose these vectors on the position plot
5. v_ave, a_ave between t = 0 and 2 s

 

Solution:

 

Video Solution

 

The following video walks you through the solution to this problem. It is suggested that you try solving the problem first and then, if you have difficulties with the solution, watch the video for help.

 

 

Interactive Solution

 

What is the velocity of the car?

 

v = (6t) i + (1.5t ²) j m/s
v = (2 + 3t ²) i + (6+ 0.5t ³) j m /s
v = (6t + 0.125t ⁴) i+ (t ³ + 2t) j m/s