A science student is riding on a flatcar of a train...

A science student is riding on a flatcar of a train traveling along a straight horizontal track at a constant speed of $10.0 \mathrm{~m} / \mathrm{s} .$ The student throws a ball into the air along a path that he judges to make an initial angle of $60.0^{\circ}$ with the horizontal and to be in line with the track. The student's professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise?

Key Concepts

Inertial Reference Frames

This concept explains that observers in different states of motion (i.e. different inertial frames) can perceive the same event differently. In the problem, the student on the moving train and the professor on the ground (stationary frame) see different trajectories for the ball even though the underlying physics remains the same.

Relative Velocity

Relative velocity involves the addition of velocities from different reference frames. The ball's velocity as seen by the professor is a combination of its velocity relative to the train and the train's velocity relative to the ground. This principle is crucial for understanding why the professor observes a different motion compared to the student on the train.

Projectile Motion

Projectile motion concerns the motion of objects under the influence of gravity, typically decomposed into horizontal and vertical components. While the student sees the ball being thrown along an angled trajectory, the analysis of the vertical component of the ball’s motion (such as calculating the maximum height reached using kinematic equations) is key to determining how high the ball rises as seen by an observer in a different frame.

Vector Decomposition

Vector decomposition breaks an initial velocity into its horizontal and vertical components. This technique is essential for solving projectile motion problems, allowing one to apply the appropriate kinematic equations separately to each component, which is especially important when the observed motion appears different in separate reference frames.

Transcript

00:01 In this exercise, we have a student inside a train, and in her reference frame, she throws a ball with an initial angle of 60 degrees above the horizontal to the left, and the initial speed of the ball is v -0.

00:18 Now, the professor, the student's professor, is outside the train on the ground, at rest, and in the professor's reference frame, the student is moving to the right with a speed of 10 meters per second.

00:32 And still in the professor's reference frame, the ball moves upward.

00:39 So what we need to calculate here is how high does the ball rise in the professor's reference frame? first of all, notice that the initial speed of the ball in the student's reference frame in the x direction is v0 times a cosine of 60.

00:58 The initial speed in the y direction is v0 times the sign of 60.

01:09 And in the professor's reference frame, i'm going to use prime to denote the speeds that the professor sees.

01:17 So the initial speed of the ball i'm going to call v0 prime.

01:22 Notice that since the professor sees the ball rising vertically, v0 prime x is zero, while v0 prime y is just v0 y.

01:43 Okay, and for that reason, the vector v0 prime is just v0 y times j.

01:52 Now notice that v0x is the horizontal speed of the ball in the professor's reference frame, and this can be written as the horizontal speed of the ball in the student's reference frame, minus the speed of the student.

02:11 And from here, we can obtain that v0 x is equal to v.

02:18 So the horizontal speed of the ball in the student's reference frame is equal to the speed of the student.

02:28 So v0 cosine of 60 is equal to v, which is 10 meters per second...

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