A train is travelling at 32 ms. The driver brakes, producing a deceleration of k716-tms after / seconds. The train comes to rest in 16 seconds. Find k, and how far the train travel before coming to rest.
To solve for k, we need to use the kinematic equation that relates velocity, initial velocity, acceleration, and time:
v = u + at
In this case, the train is traveling at an initial velocity of 32 m/s and comes to rest, so the final velocity (v) is 0 m/s. The acceleration (a) is given as k716 - t m/s^2, and the time (t) is 16 seconds.
Using the formula above, we can solve for k:
0 = 32 + (k716 - 16) * 16
0 = 32 + 16k716 - 256
16k716 = 224
k716 = 224 / 16
k = 14
Now that we have the value of k, we can find how far the train travels before coming to rest using the kinematic equation that relates displacement, initial velocity, acceleration, and time:
s = ut + (1/2)at^2
The initial velocity (u) is 32 m/s, the acceleration (a) is k716 - t m/s^2, and the time (t) is 16 seconds.
Plugging in the values, we have:
s = 32 * 16 + (1/2)(14 * 16 - 16)(16^2)
s = 512 + (1/2)(224 - 16)(256)
s = 512 + (1/2)(208)(256)
s = 512 + 54016
s = 54528
Therefore, the train travels 54528 meters before coming to rest.
To find the value of k, we can use the formula for deceleration:
acceleration = (change in velocity) / (change in time)
Given that the initial velocity of the train is 32 m/s, the final velocity is 0 m/s (since the train comes to rest), and the time taken to come to rest is 16 seconds, we can plug these values into the formula:
k716 - 0 = (0 - 32) / 16
Simplifying the equation:
k716 = -32 / 16
k716 = -2
Therefore, k = -2.
To find how far the train travels before coming to rest, we can use the equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Substituting the known values:
distance = (32 * 16) + (0.5 * -2 * 16^2)
Simplifying the equation:
distance = 512 + (-16 * 16)
distance = 512 - 256
distance = 256 meters
Therefore, the train travels 256 meters before coming to rest.
v = at + c, where a is the acceleration and c is a constant
when t = 0, v = 32, so 32 = 0 + c
v = at + 32
when t = 16, v = 0
0 = 16a + 32
a = -2
so we have
a = -2 m/s^2
v = -2t + 32
s = -t^2 + 32t + k, where s is the distance in metres
distance between t= 0 and t = 16
= (0 + 0 + k) - (-16^2 - 512 + k)
= 768