A certain rocket, initially at rest, is shot straight up...

A certain rocket, initially at rest, is shot straight up with an acceleration of $6 t$ meters per second per second during the first 10 seconds after blast-off, after which the engine cuts out and the rocket is subject only to gravitational acceleration of $-10 \mathrm{me}-$ ters per second per second. How high will the rocket go?

Key Concepts

Variable Acceleration

In kinematics, acceleration can vary with time rather than being constant. When a system has an acceleration defined as a function of time, it is crucial to recognize that the velocity of the object is the integral of the acceleration function over time. This approach is fundamental in solving problems where the force or acceleration changes over the interval of interest.

Integration in Kinematics

Integration is used to move from acceleration to velocity and from velocity to displacement. By integrating the acceleration function with the appropriate initial conditions, one obtains the velocity function; similarly, integrating the velocity function gives the displacement or position. This method is essential for analyzing motion when acceleration is not constant.

Piecewise Motion Analysis

Many real-world problems involve piecewise definitions of motion where different acceleration regimes apply over different intervals. It is necessary to solve for the motion in each phase separately and then connect them using the conditions at the transition point. This analysis ensures that the overall motion is correctly described even when the governing equations differ across time periods.

Projectile Motion under Gravity

Once the propulsion phase is over, the object typically transitions into projectile motion where the only acceleration acting is due to gravity. In such cases, the maximum height is reached when the upward velocity decreases to zero. Understanding projectile motion allows one to apply the kinematic equations with constant acceleration to determine key characteristics such as the peak altitude, time of flight, and range.

Transcript

00:01 The rocket is initially at rest.

00:04 So v0 is zero.

00:08 I'm also going to think that its initial height was also zero.

00:18 Launched from the ground.

00:21 So the acceleration is 6 meters per second for the first 10 seconds after blast off.

00:46 Okay.

00:48 So, acceleration of that for the first 10 seconds.

00:56 Okay, but you definitely have to subtract gravity, which is 9 .8 meters per second.

01:18 I think that's what it means.

01:20 Okay, so the velocity for the first 10 seconds is going to be six, well, well, 6t squared over 2, which is going to be 3t squared minus 9 .8 meters per second, which is gravity times t plus a constant.

01:45 But that constant is zero, because v0 is zero.

01:51 The distance traveled is going to be taking the integral of this, t cubed over 3.

02:01 That would be 3, just be t cubed, minus 9 .8 divided by 2 is 4 .95, i think.

02:21 9 .8 divided by 2, no, it's just 4 .9 plus a constant.

02:36 But again, this is zero.

02:40 So, this happens for 10 seconds.

02:56 10 cubed is 1 ,000.

03:02 10 squared is 100 times 4 .9 would be 490.

03:12 So it's going to be 510 meters.

03:17 Okay, after that, the engine cuts out, and the rocket is subject only to gravitational acceleration, which is negative 10.

03:31 So it's trying to tell us that gravitational acceleration is negative 10.

03:36 So if we replace this with negative 10, then we'd start at 500.