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Chapter 8: Problem 90

Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan's weight is 800 N, Jane's weight is 600 N, and that of the sleigh is 1000 N. They see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity of 5.00 m/s at 30.0\(^\circ\) above the horizontal (relative to the ice), and Jane jumps to the right at 7.00 m/s at 36.9\(^\circ\) above the horizontal (relative to the ice). Calculate the sleigh's horizontal velocity (magnitude and direction) after they jump out.

Short Answer

Expert verified
The sleigh moves left at 0.10 m/s.

Step by step solution

01

Calculate Masses from Weights

First, we need to find the masses of Jonathan, Jane, and the sleigh using the formula for weight: \( W = m \cdot g \), where \( g = 9.81 \text{ m/s}^2 \).\- For Jonathan: \( m_J = \frac{800}{9.81} \approx 81.55 \text{ kg} \).\- For Jane: \( m_j = \frac{600}{9.81} \approx 61.16 \text{ kg} \).\- For the sleigh: \( m_s = \frac{1000}{9.81} \approx 101.94 \text{ kg} \).
02

Resolve Velocities into Horizontal Components

Next, calculate the horizontal components of the velocities:\- Jonathan's horizontal velocity: \( v_{Jx} = 5.00 \cos(30.0^\circ) = 4.33 \text{ m/s} \).\- Jane's horizontal velocity: \( v_{jx} = 7.00 \cos(36.9^\circ) = 5.59 \text{ m/s} \).
03

Apply the Conservation of Momentum

Since the total momentum of the system must be conserved and initially it was zero, it remains zero after they jump. Set the equation for horizontal momentum:\[m_J v_{Jx} + m_j v_{jx} + m_s v_{sx} = 0\]Substitute the known values:\[81.55 \cdot (-4.33) + 61.16 \cdot 5.59 + 101.94 \cdot v_{sx} = 0\]Calculate the momenta and solve for \( v_{sx} \), the sleigh's velocity.
04

Solve for the Sleigh's Velocity

Simplify the momentum equation:\[-353.11 + 342.88 + 101.94 v_{sx} = 0\]Solve for \( v_{sx} \):\[v_{sx} = \frac{353.11 - 342.88}{101.94} = 0.10 \text{ m/s}\]This direction is to the left, opposite to Jonathan's jump.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frictionless Surface
When discussing physics problems involving a frictionless surface, it is essential to remember that this means the surface offers no resistance to motion. As such, if an object is already in motion, it will continue moving indefinitely at a constant speed and direction unless acted upon by another force.
A frictionless surface in this context allows us to focus on the core principles of momentum conservation without having other forces complicate the calculations.
  • This principle is beneficial when analyzing scenarios, such as Jonathan and Jane jumping off the sleigh, where we need to exclusively consider their movements and the resulting motion of the sleigh.
  • Since there is no friction, the initial horizontal momentum of Jonathan and Jane will directly affect the sleigh's motion without energy loss.
Horizontal Velocity
Understanding horizontal velocity is crucial when analyzing motion in physics. Velocity is a vector quantity meaning it has both a magnitude and a direction. In Jonathan and Jane鈥檚 case, their jumps include both vertical and horizontal components. However, we are primarily interested in the horizontal velocity as it determines the sleigh's movement along the ice.
By using trigonometric functions, we can calculate the horizontal component of each jump:
  • The equation, \( v_{x} = v \cdot \cos(\theta) \), where \( v \) is the initial velocity and \( \theta \) is the angle from the horizontal, allows us to find how much of their velocity propels them along the ice surface.
  • This separation and focus on horizontal velocity simplify calculations and make analyzing the sleigh鈥檚 motion straightforward.
Mass Calculation
Calculating mass from weight is a common step in physics problems since weight is the force resulting from a mass under gravity. Here, we use the formula \( W = m \cdot g \), where \( W \) is weight and \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity.
  • By rearranging the formula to \( m = \frac{W}{g} \), we can easily derive the masses of Jonathan, Jane, and the sleigh from their given weights.
  • These masses are then used in calculations related to momentum, allowing us to apply the conservation of momentum principle effectively.
This process ensures that the effects of gravitational force are accounted for when determining the momentum of each person as they jump off the sleigh. It highlights the relationship between weight and mass, two essential concepts in physics.

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Most popular questions from this chapter

The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A .30-caliber bullet has mass 0.00720 kg and a speed of 601 m/s relative to the muzzle when fired from a rifle that has mass 2.80 kg. The loosely held rifle recoils at a speed of 1.85 m/s relative to the earth. Find the momentum of the propellant gases in a coordinate system attached to the earth as they leave the muzzle of the rifle.

An 8.00-kg block of wood sits at the edge of a frictionless table, 2.20 m above the floor. A 0.500-kg blob of clay slides along the length of the table with a speed of 24.0 m/s, strikes the block of wood, and sticks to it. The combined object leaves the edge of the table and travels to the floor. What horizontal distance has the combined object traveled when it reaches the floor?

A 0.100-kg stone rests on a frictionless, horizontal surface. A bullet of mass 6.00 \(g\), traveling horizontally at 350 m/s, strikes the stone and rebounds horizontally at right angles to its original direction with a speed of 250 m/s. (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

Block \(B\) (mass 4.00 kg) is at rest at the edge of a smooth platform, 2.60 m above the floor. Block \(A\) (mass 2.00 kg) is sliding with a speed of 8.00 m/s along the platform toward block \(B\). \(A\) strikes \(B\) and rebounds with a speed of 2.00 m/s. The collision projects \(B\) horizontally off the platform. What is the speed of \(B\) just before it strikes the floor?

The mass of a regulation tennis ball is 57 g (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for 30 ms. (This number can also vary, depending on the racket and swing.) We shall assume a 30.0-ms contact time. The fastest-known served tennis ball was served by "Big Bill" Tilden in 1931, and its speed was measured to be 73 m/s. (a) What impulse and what force did Big Bill exert on the tennis ball in his record serve? (b) If Big Bill's opponent returned his serve with a speed of 55 m/s, what force and what impulse did he exert on the ball, assuming only horizontal motion?
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