A driver in a car traveling at a speed of 56.0 mi/h sees a deer 116 m away on the road. Calculate the minimum constant acceleration that is necessary for the car to stop without hitting the deer (assuming that the deer does not move in the meantime).
change mi/hr to m/s
vf^2=vi^2+2ad solve for a.
To calculate the minimum constant acceleration required to stop the car without hitting the deer, we can use the equations of motion. Let's break down the problem step by step:
Step 1: Convert the speed to meters per second (m/s)
We need to convert the speed of the car from miles per hour to meters per second because the distance is given in meters. The conversion factor is 1 mile = 1609.34 meters, and 1 hour = 3600 seconds.
Speed in m/s = (Speed in mi/h) * (1609.34 m/1 mi) * (1 hr/3600 s)
Speed in m/s = 56.0 * (1609.34/3600)
Speed in m/s = 25.02 m/s (rounded to two decimal places)
Step 2: Define the known variables
Let's assign variables to the known values in the problem:
- Initial velocity (u) = 25.02 m/s
- Final velocity (v) = 0 m/s (the car needs to stop)
- Initial position (s) = 0 m (starting point)
- Final position (s') = 116 m
Step 3: Apply the equation of motion
We will use the equation of motion that relates the final velocity, initial velocity, acceleration, and displacement.
v^2 = u^2 + 2a(s' - s)
Here, v is the final velocity, u is the initial velocity, a is the acceleration, s' is the final position, and s is the initial position.
Plugging in the known values:
0^2 = (25.02)^2 + 2a(116 - 0)
Simplifying the equation:
0 = 625.2004 + 232a
Step 4: Solve for acceleration (a)
To find the acceleration (a), we rearrange the equation and solve for a:
a = -625.2004 / 232
a ≈ -2.69 m/s^2 (rounded to two decimal places)
Step 5: Interpret the result
The minimum constant acceleration that is necessary for the car to stop without hitting the deer is approximately -2.69 m/s^2. The negative sign indicates that the acceleration is in the opposite direction of the initial velocity, which means the car needs to decelerate.