A car traveling at a constant speed of 33 m/s passes a trooper hidden behind a billboard. One second later the trooper starts the car with a constant acceleration of 2.62 m/s^2.
How long after the trooper starts the chase does he overtake the speeding car? (answer in units of s)
distance car= 33m/s * time
distance cop= 1/2 a (t-1)^2
the distances are the same, set the two equal,and solve for t. You may need the quadratic equation.
To find the time it takes for the trooper to overtake the speeding car, we need to determine the time it takes for the trooper to match the speed of the car. This is the point at which the trooper catches up with the car.
Let's break down the problem step-by-step:
Step 1: Determine the initial velocity of the trooper's car.
Since the trooper's car was initially at rest, the initial velocity (v₀) is 0 m/s.
Step 2: Calculate the distance traveled by the car during the time the trooper starts the chase.
Using the formula: distance = velocity × time, we can calculate the distance traveled by the car. Since the car has a constant speed of 33 m/s and the trooper starts one second later, the time (t) for the car is one second less than the time for the trooper.
distance_car = velocity_car × time_car
= 33 m/s × (t - 1 s)
Step 3: Determine the distance traveled by the trooper's car during the time it takes to match the speed of the car.
Using the second equation of motion: distance = initial velocity × time + 0.5 × acceleration × time², we can calculate the distance traveled by the trooper's car.
distance_trooper = initial velocity × time_trooper + 0.5 × acceleration_trooper × time_trooper²
= 0 m/s × t + 0.5 × 2.62 m/s² × t²
= 1.31 m/s² × t²
Step 4: Set the distances traveled by the car and the trooper's car equal to each other.
Since the trooper catches up with the car, the distances traveled by both vehicles are the same.
distance_car = distance_trooper
33 m/s × (t - 1 s) = 1.31 m/s² × t²
Step 5: Rearrange the equation and solve for t.
We need to solve the quadratic equation to find the time it takes for the trooper to overtake the car.
1.31 m/s² × t² - 33 m/s × t + 33 m/s = 0
Using the quadratic formula: t = (-b ± √(b² - 4ac)) / (2a), where a = 1.31 m/s², b = -33 m/s, and c = 33 m/s.
t = (-(-33) ± √((-33)² - 4 × 1.31 × 33)) / (2 × 1.31)
Simplifying the equation gives us two values for t: t₁ and t₂.
Step 6: Select the positive value of t as the answer since time cannot be negative.
The trooper cannot overtake the car in negative time, so we choose the positive value of t.
t = t₁
This positive value of t represents the time it takes for the trooper to overtake the speeding car. The answer is in units of seconds.