Background

Because the marble moved both horizontally (farther from the table) and vertically (closer to the ground), we say that we are dealing with a trajectory, because the marble’s motion is also influenced by gravity. Other examples of objects moving in a trajectory would be:

a football moving mid-aira heavy object being aimed at a target using a catapult

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Analyze

As shown in Figure 1, we are given two values: the initial horizontal velocity of the marble (1.50 m/s) and its distance from the table after falling to the ground (0.70 m). What we are asked is the velocity of the marble after it hits the ground, which is its final velocity.

However, we know that the marble moves in two dimensions: horizontal and vertical. Therefore, we not only have velocity in the horizontal direction, but also in the vertical direction (see Figure 2). As a result, the marble can be thought of as moving:

Horizontally, at a constant

Vertically, from rest, and speeding up by

because of gravityAssumption

For this problem, we will assume that air resistance is negligible. (To learn how it affects motion, see

Solve

Let's list down all our knowns and unknowns. We denote them as the following:

Initial horizontal velocity of the marble:

Initial vertical velocity of the marble:

Vertical acceleration due to gravity:

Horizontal distance of marble from the table:

Final velocity of the marble:

To solve the problem, we break the problem down to three smaller steps.

Time it took for the marble to reach the ground

This will be easy because we are given the initial horizontal velocity (which does not change) and the horizontal distance travelled by the marble. This represents the horizontal component of our trajectory.

It took the marble 0.4667 s to reach the ground.

Final vertical velocity of the marble

It will also take 0.4667 s for the marble to reach the ground if we were to drop it from the height of the table. This represents the vertical component of our trajectory.

The final vertical velocity of the marble was -4.578 m/s, which implies that the velocity is downwards.

Final velocity of the marble

The question did not state whether it's the horizontal or vertical velocity that's needed. Therefore, we should assume that the question asks for the vector sum of the horizontal and vertical velocities. (Please see Figure 3.) The magnitude of the final velocity can be calculated as:

The horizontal velocity does not change, so . Hence,

The final speed of the marble just before it hits the ground is therefore around 4.82 m/s.

Meanwhile, the direction of the final speed can be calculated by noting that and form a right triangle (explained here ). Therefore, the direction, represented by , can be calculated using the arctangent:

Therefore, the velocity of the marble just before it hits the ground is 4.82 m/s, -71.86° from the horizontal.