A police radar has an effective range of 1.0 km, and a motorist's radar detector has a range of 1.9 km. The motorist is going 100km/h in a 80km/h zone when the radar detector beeps. At what rate must the motorist decelerate to avoid a speeding ticket?
Find: a
Given:
s = (1.9-1.0)[km]
u = 100[km/h]
v = 80[km/h]
Using two equations:
<1>: s = u t + ½ a t²
<2>: v = u + a t
Rearrange <2> to get t.
t = (v-u)/a
Substitute this into <1> to get a.
s = u(v-u)/a + (v-u)²/(2a)
Rearrange
.: a = (v²-u²)/(2s)
Calculate using the given values.
I suggest converting from [km/h²] to [m/s²]
34
2000
To find the rate at which the motorist must decelerate to avoid a speeding ticket, we need to determine the time it will take for the motorist to reach the effective range of the police radar.
Let's start by converting the speeds from km/h to m/s for consistency.
The motorist's speed is 100 km/h, which is approximately (100 * 1000) m/3600 s = 27.78 m/s.
The speed limit in the 80 km/h zone is 80 km/h, which is approximately (80 * 1000) m/3600 s = 22.22 m/s.
Next, let's determine the time it will take for the motorist to reach the effective range of the police radar.
The range of the motorist's radar detector is 1.9 km, which is (1.9 * 1000) m = 1900 m.
The range the motorist has to cover before the radar detector beeps is 1900 m - 1000 m = 900 m (assuming the motorist detects the radar signal right at the edge of the police radar's range).
The time it will take for the motorist to cover this distance is given by the equation: time = distance / speed. Therefore, time = 900 m / (27.78 m/s) ≈ 32.41 s.
Now, let's find the deceleration rate needed to avoid a speeding ticket.
The initial velocity of the motorist is 27.78 m/s, and the final velocity after deceleration will be 22.22 m/s (the speed limit).
The deceleration rate can be calculated using the equation: acceleration = (final velocity - initial velocity) / time.
Therefore, acceleration = (22.22 m/s - 27.78 m/s) / 32.41 s ≈ -0.17 m/s^2.
Note that the negative sign indicates deceleration or slowing down.
Therefore, the motorist must decelerate at a rate of approximately 0.17 m/s^2 to avoid a speeding ticket.