An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at +13.0 m/s^2. At t1 the rocket shut down and the sled moves with constant velocity v until t2. The total distance traveled by the sled is 5.30 x 10^3m and the total time is 90s. Find t1, t2, and v.
Write an equation for distance travelled (X) vs times t1 and t2.
Call it X(t1,t2) = 5300 m
v = 13.0 t1
You also know that t1 + t2 = 90
Solve the three equations in three unkowns.
To solve this problem, we can break it down into two parts: the time it took for the sled to accelerate and the time it took for the sled to move with constant velocity.
Let's start with the time it took for the sled to accelerate (t1):
Using the kinematic equation: d = v0*t + (1/2)*a*t^2
where
d = distance
v0 = initial velocity
t = time
a = acceleration
Since the sled starts from rest (v0 = 0) and the distance traveled during acceleration is not given, we can use the equation in a simplified form:
d = (1/2)*a*t^2
Substituting the known values:
5.30 x 10^3m = (1/2)*(+13.0 m/s^2)*t1^2
Now we can solve for t1:
10.6 x 10^3m = 13.0 m/s^2 * t1^2
Divide both sides by 13.0 m/s^2:
t1^2 = (10.6 x 10^3m) / (13.0 m/s^2)
t1^2 = 815.4 s^2
Take the square root of both sides to find t1:
t1 = √(815.4 s^2)
t1 ≈ 28.57 s (rounded to two decimal places)
Now that we have t1, let's find t2:
Total time (90 s) is the sum of t1 and t2:
90 s = t1 + t2
Rearrange the equation to solve for t2:
t2 = 90 s - t1
t2 = 90 s - 28.57 s
t2 ≈ 61.43 s
Finally, let's find the constant velocity (v) using the equation:
Total distance traveled (5.30 x 10^3m) is the sum of the distance during acceleration and the distance during constant velocity:
5.30 x 10^3m = (1/2)*(+13.0 m/s^2)*t1^2 + v * t2
Substituting the known values:
5.30 x 10^3m = (1/2)*(+13.0 m/s^2)*(28.57 s)^2 + v * 61.43 s
Now we can solve for v:
5.30 x 10^3m = 26.0 m/s^2 * (815.4 s^2) + v * 61.43 s
Subtract 26.0 m/s^2 * (815.4 s^2) from both sides:
5.30 x 10^3m - 26.0 m/s^2 * (815.4 s^2) = v * 61.43 s
Calculate the left-hand side:
5.30 x 10^3m - 21,204 m^2/s^2 ≈ -15,904 m^2/s^2
Divide both sides by 61.43 s:
v ≈ (-15,904 m^2/s^2) / (61.43 s)
v ≈ - 259 m/s (rounded to three decimal places)
Therefore, t1 ≈ 28.57 s, t2 ≈ 61.43 s, and v ≈ - 259 m/s.