A car traveling at a constant speed of 39.6 m/s passes a trooper hidden behind a billboard. One second later the trooper starts the car with a constant acceleration of 2.25 m/s^2

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How long after the trooper starts the chase does he overtake the speeding car?
Answer in units of s

Write equations for the distance travelled by each car after the car passes at t = 0. Set the two distances equal and solve for t.

X1 = 39.6 t
X2 = (2.25/2)(t-1)^2

X1 = X2

Solve for t. The time they are asking for is t-1.

To find the time it takes for the trooper to overtake the speeding car, we need to determine the time at which both vehicles are at the same position. Let's break down the problem into two parts:

1. Calculate the position function for the car: Since the car is traveling at a constant speed, its position function can be described as s_car(t) = v_car * t, where v_car is the speed of the car and t is the time.
Let's plug in the given values: s_car(t) = 39.6 m/s * t.

2. Calculate the position function for the trooper: Since the trooper starts from rest and accelerates at a constant rate, we can use the equation of motion s_trooper(t) = u_trooper * t + (1/2) * a_trooper * t^2, where u_trooper is the initial velocity of the trooper, a_trooper is the acceleration of the trooper, and t is the time.
Substituting in the given values: s_trooper(t) = 0 * t + (1/2) * 2.25 m/s^2 * t^2.

Now, we need to find the time when the trooper overtakes the car. This occurs when the position of the trooper equals the position of the car. Therefore, we can set the two position functions equal to each other and solve for t:

39.6 m/s * t = (1/2) * 2.25 m/s^2 * t^2.

Simplifying the equation:

39.6 * t = 1.125 * t^2.

Rearranging the equation to one side:

1.125 * t^2 - 39.6 * t = 0.

Factoring out t:

t * (1.125 * t - 39.6) = 0.

This equation can be satisfied if either t = 0 (which is not applicable in this case) or (1.125 * t - 39.6) = 0.

Solving for t:

1.125 * t = 39.6.

Dividing both sides by 1.125:

t = 39.6 / 1.125.

Calculating the value:

t = 35.2 seconds.

Therefore, the trooper overtakes the car 35.2 seconds after starting the chase.