A man driving his car into 20.0 ft garage with a velocity of 20.0mi/h applies the brakes, producing a constant deceleration. Find the smallest deceleration necessary to avoid striking the back wall of the garage. And find how many seconds it takes for the car to come to rest
v = Vi + a t
v = 0 after 20 ft we hope
Vi = 20 * 5280ft/mi /3600s/h = 29.33 ft/s
so
a t = -29.33
a = -29.33/t
d = Vi t + (1/2) a t^2
20 = 29.33 t + .5 (-29.33/t)(t^2)
20 = .5 * 29.33 t
t = 1.36 seconds
a = - 21.6 ft/s^2
To find the smallest deceleration necessary to avoid striking the back wall of the garage, we can use the kinematic equation that relates distance, initial velocity, final velocity, and acceleration:
distance = (final velocity² - initial velocity²) / (2 * acceleration)
In this case, the distance is given as 20.0 ft, the initial velocity is 20.0 mi/h, and we need to find the acceleration.
First, we need to convert the initial velocity from miles per hour (mi/h) to feet per second (ft/s). Since 1 mile = 5280 feet and 1 hour = 3600 seconds, we have:
initial velocity = 20.0 mi/h * (5280 ft/mi) / (1 h/3600 s) = 29.3 ft/s
Next, we substitute the values into the equation and solve for acceleration:
20.0 ft = (0 - 29.3 ft/s)² / (2 * acceleration)
Rearranging the equation, we get:
acceleration = (29.3 ft/s)² / (2 * 20.0 ft)
Simplifying,
acceleration = 42.87 ft/s²
Therefore, the smallest deceleration necessary to avoid striking the back wall of the garage is approximately 42.87 ft/s².
To find how many seconds it takes for the car to come to rest, we can use another kinematic equation that relates final velocity, initial velocity, acceleration, and time:
final velocity = initial velocity + (acceleration * time)
In this case, the final velocity is 0 (since the car comes to rest), the initial velocity is 29.3 ft/s, and we need to find the time.
Rearranging the equation, we get:
time = (final velocity - initial velocity) / acceleration
Substituting the values,
time = (0 - 29.3 ft/s) / 42.87 ft/s²
Simplifying,
time ≈ -0.682 seconds
However, time cannot be negative in this context. Therefore, we can conclude that it takes approximately 0.682 seconds for the car to come to rest.