A car travelling 19.4m/s passes apolice car at rest.As it passes, the police car starts up, accelerating with a magnitude of 3.2m/s. Maintaining that acceleration, how long will it take the police car to catch up with the speeding motorist?
vt=at²/2
t=2v/a=2•19.4/3.2=121.25 s=2.02 min
To determine how long it will take the police car to catch up with the speeding motorist, we can use the equations of motion.
Let's assume that after time t, both vehicles have traveled the same distance d.
For the speeding car:
Distance traveled, d1 = 19.4t
For the accelerating police car:
Distance traveled, d2 = (1/2) * a * t^2
Where:
d1= Distance traveled by the speeding car
d2 = Distance traveled by the police car
a = Acceleration of the police car
t = Time taken
Since both vehicles have traveled the same distance when the police car catches up, we can equate the two equations:
19.4t = (1/2) * 3.2 * t^2
Let's solve for t:
19.4t = 1.6t^2
Divide both sides by t:
19.4 = 1.6t
Divide both sides by 1.6:
t = 19.4 / 1.6
t ≈ 12.125 seconds
So, it will take the police car approximately 12.125 seconds to catch up with the speeding motorist.
To find the time it takes for the police car to catch up with the speeding motorist, we can use the equation of motion:
Distance = Initial velocity × Time + 0.5 × Acceleration × Time^2
For the motorist:
Initial velocity = 19.4 m/s
Distance = Unknown (we'll call it D)
Acceleration = 0 m/s^2 (since the motorist maintains a constant velocity)
For the police car:
Initial velocity = 0 m/s (at rest)
Acceleration = 3.2 m/s^2
Distance = D
When the police car catches up to the motorist, they will have traveled the same distance. Therefore, we can set up the equation for both the motorist and police car, and solve for time.
For the motorist:
Distance = 19.4t
For the police car:
Distance = 0.5 × 3.2t^2
Since both distances are equal, we can write the equation:
19.4t = 0.5 × 3.2t^2
Simplifying this equation, we get:
9.7t = 1.6t^2
Dividing both sides by t, we get:
9.7 = 1.6t
Now, we can solve for t by dividing both sides by 1.6:
t = 9.7 / 1.6
t ≈ 6.06 seconds
Therefore, it will take approximately 6.06 seconds for the police car to catch up with the speeding motorist.