Same rental cars have a GPS unit installed, which allows the rental car company to check where you are at all times and thus also know your speed at any time. One of these rental cars is driven by an employee in the company's lot and during the time interval from 0 to 10s, is found to have a position vector as a function of time of
r→(t) =( (24.4m)-t(12.3m/s)+t²(2.43m/s²),
(74.4m+t²(1.80m/s²)-t³(0.130m/s³) )
a) what is the distance of this car from the origin of the coordinate system at t=5.00s?
b) what is the velocity vector as a function of time?
c) what is the speed at t=5.00s?
Extra credit: Can you produce a plot of the trajectory of the car in the xy-plane?
I need a help ...
a) To find the distance of the car from the origin at t=5.00s, we need to find the magnitude of the position vector at that time.
The position vector at t=5.00s is:
r→(5.00s) =( (24.4m)-(5.00s)(12.3m/s)+(5.00s)²(2.43m/s²), (74.4m+(5.00s)²(1.80m/s²)-(5.00s)³(0.130m/s³) )
We can simplify this expression to calculate the magnitude:
r→(5.00s) =(24.4m)-(5.00s)(12.3m/s)+(5.00s)²(2.43m/s²)
+(74.4m+(5.00s)²(1.80m/s²)-(5.00s)³(0.130m/s³) )
Now let's calculate the magnitude of this expression:
| r→(5.00s) | = √(x² + y²)
Where x and y are the x-component and y-component of the position vector, respectively.
|x| = | (24.4m)-(5.00s)(12.3m/s)+(5.00s)²(2.43m/s²) |
|y| = | (74.4m+(5.00s)²(1.80m/s²)-(5.00s)³(0.130m/s³) ) |
Now, substitute t=5.00s into these expressions and calculate each component separately. Finally, find the magnitude using the formula mentioned above.
b) To find the velocity vector as a function of time, we need to take the derivative of the position vector with respect to time.
Velocity vector, v→(t) = dr→(t)/dt
To find the derivative, we differentiate each component of the position vector with respect to time.
vx = (d/dt) ( (24.4m)-(t)(12.3m/s)+(t²)(2.43m/s²) )
vy = (d/dt) ( (74.4m+t²(1.80m/s²)-t³(0.130m/s³) )
Differentiate each component separately to get the velocity vector as a function of time.
c) To find the speed at t=5.00s, we need to calculate the magnitude of the velocity vector at that time.
Speed, |v→(5.00s)| = √( vx² + vy² )
Calculate the components vx and vy using the derivative expressions obtained in part (b). Substitute t=5.00s and calculate the magnitude using the formula mentioned above.
Extra credit: To produce a plot of the trajectory of the car in the xy-plane, we need to plot the x and y positions of the car as a function of time.
On the x-axis, we plot the x component of the position vector (x = (24.4m)-(t)(12.3m/s)+(t²)(2.43m/s²)). On the y-axis, we plot the y component of the position vector (y = (74.4m+t²(1.80m/s²)-t³(0.130m/s³) )).
For each value of t, calculate the corresponding x and y values using the given expressions. Then plot these points on a graph with time on the x-axis and position (x, y) on the y-axis. Connect the points to visualize the trajectory of the car in the xy-plane.