An 83.0 -kg running back leaps straight ahead toward the end zone with a speed of \(6.50 \mathrm{~m} / \mathrm{s}\). A 115 -kg linebacker, keeping his feet on the ground, catches the running back and applies a force of \(900 . \mathrm{N}\) in the opposite direction for 0.750 s before the running back's feet touch the ground. a) What is the impulse that the linebacker imparts to the running back? b) What change in the running back's momentum does the impulse produce? c) What is the running back's momentum when his feet touch the ground? d) If the linebacker keeps applying the same force after the running back's feet have touched the ground, is this still the only force acting to change the running back's momentum?

Step by step solution

01

(Step 1: Calculate the initial momentum of the running back)

First, we can find the initial momentum of the running back using the formula: \(momentum = mass \times velocity\). With \(mass=83.0\,\text{kg}\) and \(velocity=6.50\,\text{m/s}\): \(initial\_momentum = 83.0 \,\text{kg} \times 6.50\,\text{m/s} = 539.5\,\text{kg}\cdot\text{m/s}\)

02

(Step 2: Calculate the impulse delivered by the linebacker)

Impulse is defined as the product of force and time: \(Impulse = Force \times Time\). In this case, we have \(force = -900\,\text{N}\) (negative sign as it acts in the opposite direction of the running back's motion), and \(time = 0.750\,\text{s}\). Calculate the impulse: \(Impulse = -900\,\text{N} \times 0.750\,\text{s} = -675\,\text{kg}\cdot\text{m/s}\) The impulse that the linebacker imparts to the running back is -675 kg.m/s.

03

(Step 3: Determine the change in momentum of the running back)

The change in momentum is equal to the impulse delivered by the linebacker: \(\Delta momentum = Impulse\) Thus, the change in the running back's momentum is -675 kg.m/s.

04

(Step 4: Calculate the final momentum of the running back)

We know that the final momentum of the running back is the sum of his initial momentum and the change in momentum (impulse provided by the linebacker): \(final\_momentum = initial\_momentum + \Delta momentum\) \(final\_momentum = 539.5\,\text{kg}\cdot\text{m/s} - 675\,\text{kg}\cdot\text{m/s} = -135.5\,\text{kg}\cdot\text{m/s}\) The running back's momentum when his feet touch the ground is -135.5 kg.m/s.

05

(Step 5: Determine the forces acting on the running back when his feet touch the ground)

When the running back's feet touch the ground, the force of gravity (weight) also acts on him, along with the opposing force applied by the linebacker. The force of gravity acts vertically downward while the linebacker's force still acts horizontally. Therefore, the linebacker's force is not the only force acting on the running back when his feet have touched the ground.

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