Question

You (mass 55 kg) are riding a frictionless skateboard (mass 5.0 kg) in a straight line at a speed of 4.5 m/s. A friend standing on a balcony above you drops a 2.5-kg sack of flour straight down into your arms. (a) What is your new speed while you hold the sack? (b) Since the sack was dropped vertically, how can it affect your \(horizontal\) motion? Explain. (c) Now you try to rid yourself of the extra weight by throwing the sack straight up. What will be your speed while the sack is in the air? Explain.

Short answer

(a) The new speed is 4.32 m/s. (b) The sack does not affect horizontal motion because momentum is conserved separately in vertical and horizontal directions. (c) Your speed remains 4.32 m/s while the sack is in the air.

Step-by-step solution

Understand the Conservation of Momentum

Since the system is isolated and frictionless, we will use the conservation of momentum to determine the velocity after catching the sack. The formula for momentum is given by \( p = mv \), where \( m \) is mass and \( v \) is velocity. Initially, you and the skateboard are moving together, and the sack has no horizontal momentum because it is falling vertically.

Calculate Initial Momentum

Initial momentum (before catching the sack) is calculated by multiplying the total mass of you and the skateboard by your initial velocity. \[ p_{initial} = (55 \, \text{kg} + 5.0 \, \text{kg}) \times 4.5 \, \text{m/s} \] \[ p_{initial} = 60 \, \text{kg} \times 4.5 \, \text{m/s} = 270 \, \text{kg} \cdot \text{m/s} \]

Calculate Final Momentum

After catching the sack, you, the skateboard, and the sack now have a combined mass. The final momentum must equal the initial momentum because no external horizontal forces are acting. \[ p_{final} = (55 \, \text{kg} + 5.0 \, \text{kg} + 2.5 \, \text{kg}) \times v_{final} \]

Solve for Final Velocity

Set the final momentum equal to the initial momentum to solve for the new speed \( v_{final} \). \[ 270 \, \text{kg} \cdot \text{m/s} = 62.5 \, \text{kg} \times v_{final} \] \[ v_{final} = \frac{270 \, \text{kg} \cdot \text{m/s}}{62.5 \, \text{kg}} = 4.32 \, \text{m/s} \]

Understand the Vertical Motion Impact

The vertical drop of the sack does not directly affect the horizontal momentum because momentum is a vector quantity and is conserved separately in each direction. The sack was only imparted vertical momentum when dropped, but it joined your existing horizontal motion when caught.

Calculate Velocity after Throwing the Sack

When throwing the sack straight up, only vertical momentum is influenced. The horizontal momentum remains constant because there is still no external horizontal force. Therefore, your horizontal speed remains \( 4.32 \, \text{m/s} \) while the sack is in the air.

Key concepts

Momentum

Momentum is a fundamental concept in physics that measures the motion of an object. It is essentially the product of an object's mass and velocity, expressed as \( p = mv \). This means that momentum depends on both how much mass an object has and how fast it is moving.

• For instance, a heavier object moving at the same speed as a lighter one will have more momentum.
• Likewise, an object moving faster will have more momentum than the same object moving slower.

In the context of this exercise, when a sack of flour is dropped vertically onto you while on a skateboard, the principle of conservation of momentum plays a crucial role. The system (you, the skateboard, and the sack) should conserve its total momentum in the absence of external forces, particularly in the horizontal direction. This means that the total horizontal momentum before and after catching the sack remains the same.
Therefore, when the sack joins your motion, it does not add to the horizontal momentum but redistributes the existing momentum over a now larger mass (your mass plus the sack's). This helps us compute the new velocity after catching the sack.

Horizontal Motion

Horizontal motion refers to any motion happening parallel to the horizon. Understanding how forces affect horizontal motion helps us predict how objects move.
In physics, horizontal motion is especially crucial when dealing with multiple dimensions, as each direction can be analyzed separately. In this problem, since the sack is dropped vertically, it initially carries no horizontal momentum. However, once you catch it, it becomes part of the moving system.

• Even though it was dropped, its presence does not change the total horizontal momentum of the system.
• This is because horizontal and vertical motions have independent momentum."

Therefore, catching the sack doesn't change the horizontal speed unless an external force acts on the system. This concept, known as decoupled motion, is vital for understanding two- or three-dimensional problems in physics.

Velocity

Velocity is the measure of how fast an object moves in a specific direction. It’s different from speed, which only tells us how fast something is moving regardless of direction.
In the exercise, your initial velocity was 4.5 m/s. After catching the sack of flour, you and your skateboard momentarily move together with the sack at a new velocity because of the added mass.

• The velocity can be updated using the principle of conservation of momentum, indicating no external horizontal forces are at play.
• The new velocity, calculated using the initial and final masses, comes out to 4.32 m/s.

Interestingly, when you throw the sack straight up, your horizontal velocity does not change (stays at 4.32 m/s) since the momentum in the horizontal direction remains unaffected. This is because the act of throwing the sack impacts only the vertical axis. In physics, such considerations help us understand object trajectories and motion behavior in detail.