A police car is travelling in a straight line with constant speed vp.A truck travelling in the same direction with speed 3/2 vp passes the police car.The truck driver realizes that she is speeding and immediately begins to slow down at a constant rate until she comes to a stop. This is her lucky day, however, and the (still moving with the same constant speed) passes the truck driver without giving her ticket. a) Show that the truck¡¯s speed at the instant that the police car passes the truck does not depend on the magnitude of the truck¡¯s acceleration as it slows down and find the value of that speed.b) Sketch the x-t graph for the two vehicles.
To solve this problem, we can use the concept of relative velocity. Let's break it down step by step.
a) We need to show that the truck's speed at the instant the police car passes does not depend on the magnitude of the truck's acceleration.
Let's assume that the initial speed of the truck is vt and its acceleration is a. Since the truck is slowing down, the acceleration will be negative.
At the moment the police car passes the truck, both vehicles have traveled the same distance. Let's call this distance d.
The time taken by the police car to reach the point where it passes the truck is given by:
Time taken by police car = d / vp ("d" divided by the speed of the police car)
Now let's consider the truck. To find its speed at the instant the police car passes, we need to find the time it takes for the truck to reach the same point.
The equation of motion for the truck is:
d = (vt) * t + (1/2) * a * t^2
Here, "t" is the time taken by the truck to reach the point where the police car passes it.
Since the truck starts from rest, its initial speed vt is zero. Substituting the values, we get:
d = (1/2) * a * t^2
Rearranging the equation, we can find the time taken by the truck:
t = sqrt((2 * d) / a)
Now, we can find the speed of the truck at this instant:
Speed of truck = vt + a * t (Final velocity formula)
Since vt is zero, the equation simplifies to:
Speed of truck = a * t
Replacing the value of t, we get:
Speed of truck = a * sqrt((2 * d) / a) = sqrt(2 * d * a)
Therefore, the speed of the truck at the instant the police car passes it is independent of the magnitude of the truck's acceleration and is given by sqrt(2 * d * a).
b) To sketch the x-t graph for the two vehicles, we need to consider their positions as functions of time.
For the police car, its position (x) is given by:
x = vp * t (Since it travels with a constant speed vp)
For the truck, its position (x) is given by:
x = vt * t + (1/2) * a * t^2 (Equation of motion for the truck)
We can plot these two functions on the same graph, with time (t) on the x-axis and position (x) on the y-axis. The x-t graph would have a straight line for the police car and a curved line for the truck.
Note: The actual shape of the graph would depend on the specific values of vp, vt, and a.