91Ó°ÊÓ

In physics, momentum is a crucial concept broadly defined as the product of an object's mass and velocity. Dan and his skateboard are moving together at the beginning of the scenario. The momentum for both must be calculated by combining their individual momenta.

To determine the initial total momentum, we apply the formula \( p = m \times v \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. Here, calculate the momentum for Dan and the skateboard, then add these together since they are initially moving at the same speed of \(4.0 \ m/s\).

For Dan, the momentum is \( 50 \, \mathrm{kg} \times 4.0 \, \mathrm{m/s} = 200 \, \mathrm{kg \cdot m/s} \). For the skateboard, it’s \( 5.0 \, \mathrm{kg} \times 4.0 \, \mathrm{m/s} = 20 \, \mathrm{kg \cdot m/s} \). Thus, the initial total momentum is:

• Dan’s momentum = 200 \( \mathrm{kg \cdot m/s} \)
• Skateboard’s momentum = 20 \( \mathrm{kg \cdot m/s} \)
• Total Initial Momentum: \( 220 \, \mathrm{kg \cdot m/s} \)

This total of \(220 \, \mathrm{kg \cdot m/s} \) sets the stage for understanding the changes after Dan jumps off.

After Dan jumps backward off the skateboard, the situation changes. The skateboard shoots forward with a speed of \(8.0 \, \mathrm{m/s}\), and we need to determine its final momentum.

Applying the same momentum formula, the final momentum of the skateboard can be found by multiplying its mass by its new velocity:

\( 5.0 \, \mathrm{kg} \times 8.0 \, \mathrm{m/s} = 40 \, \mathrm{kg \cdot m/s} \).

• Skateboard's Final Momentum: \( 40 \, \mathrm{kg \cdot m/s} \)

Understanding this step is key because momentum conservation tells us the total momentum of the system before and after the event must be equal if no external forces act. This final momentum calculation will be used to find Dan's final velocity after he jumps off.

Now, we move to the exciting part: finding the speed at which Dan's feet hit the ground. Thanks to the principle of momentum conservation, we know that Dan and the skateboard's total initial momentum should equal the total final momentum of both, after Dan has jumped off.

To find Dan's final momentum, we subtract the skateboard's final momentum from the initial total momentum:

\( 220 \, \mathrm{kg \cdot m/s} - 40 \, \mathrm{kg \cdot m/s} = 180 \, \mathrm{kg \cdot m/s} \).

We then use the formula \( v = \frac{p}{m} \) (velocity equals momentum divided by mass) to find his velocity:

\( v = \frac{180 \, \mathrm{kg \cdot m/s}}{50 \, \mathrm{kg}} = 3.6 \, \mathrm{m/s} \).

• Dan's Final Velocity: \( 3.6 \, \mathrm{m/s} \), in the opposite direction of the skateboard.

Thus, Dan travels backward at \( 3.6 \, \mathrm{m/s} \) after jumping, illustrating both the perfection of momentum conservation and the importance of understanding each quantity involved!