The Spent Rocket Parts: During launches, rockets often discard parts. A certain rocket starts from rest on the launch pad and accelerates upward at a steady 3.3 m/s2. When it is 235 m above the launch pad, the rocket discards a used fuel canister by simply disconnecting it. Once it is disconnected the only force acting on the canister is gravity.
a.How high is the rocket when the canister hits the launch pad, assuming the rocket does not change its acceleration?
b.What total distance did the canister travel between its release and its crash onto the launch pad?
To solve this problem, we need to apply the equations of motion and consider the motion of both the rocket and the discarded fuel canister separately. Since the rocket and the fuel canister are separated after the canister is discarded, we can analyze their motions independently.
a. To find how high the rocket is when the canister hits the launch pad, we need to determine the time it takes for the canister to fall from its release point to the ground.
First, we find the time it takes for the canister to reach the ground by using the equation for free fall motion:
h = (1/2)gt^2
where h is the height (235 m), g is the acceleration due to gravity (-9.8 m/s^2), and t is the time.
Rearranging the equation to solve for t:
t = sqrt(2h / g)
Substituting the given values:
t = sqrt(2 * 235 / 9.8) = 7.67 seconds
Now, we need to find the height of the rocket at this time. To do this, we use the equation for uniform acceleration:
h = ut + (1/2)at^2
where h is the height, u is the initial velocity (0 m/s), a is the acceleration (3.3 m/s^2), and t is the time.
Rearranging the equation to solve for h:
h = (1/2)at^2
Substituting the given values:
h = (1/2) * 3.3 * (7.67^2) = 114.64 meters
Therefore, the rocket reaches a height of approximately 114.64 meters when the canister hits the launch pad.
b. To find the total distance traveled by the canister between its release and its crash onto the launch pad, we can calculate the distance covered by the canister during free fall.
The equation for the distance covered during free fall is:
d = (1/2)gt^2
where d is the distance, g is the acceleration due to gravity (-9.8 m/s^2), and t is the time.
Substituting the known values:
d = (1/2) * (-9.8) * (7.67^2) = 286.99 meters
Therefore, the canister travels a total distance of approximately 286.99 meters from its release point to its crash onto the launch pad.