A 65 kg dancer leaps .32 m high.

a) With what momentum does the dancer reach the ground?

b) What impulse is needed to make a stop?

c) As the dancer lands, the knees bend, lengthening the stopping time to .050 s. Find the average force exerted on the body.

d) Compare the stopping force to the performer's weight.

Thoughts: Momentum equals mass multiplied by the change in velocity. Impulse is the force multiplied by time.

a) Use energy conservation considerations to compute the takeoff velocity.

(1/2) M V^2 = M g H
V = sqrt (2gH)
The landing velocity will be the same. Multiply that V by M for the momentum
(b) Same number as (a)
(c) Force = Impluse/Time
(d) Compare (c) answer to Mg

A) Momentum = (2.465)(65) = 160.222 kgm/s

B) 160.222 kgm/s
C) Force = 160.222/.050 = 3204.441 N
D) 3204.441/[(65)(9.8)] = 5.031

a) Well, the momentum of the dancer can be found using the equation momentum = mass × velocity. Since the dancer is leaping straight up, we know that the initial vertical velocity is 0. So, the momentum just before reaching the ground would also be 0.

b) To make a stop, we need to apply an impulse that would completely cancel out the dancer's momentum. The impulse is given by the equation impulse = change in momentum. As we found earlier, the momentum is 0, so the impulse needed to stop the dancer would also be 0. Easy peasy, no impulse needed!

c) Ah, but wait! As the dancer lands, the knees bend, increasing the stopping time. To find the average force exerted on the body, we can use the equation average force = change in momentum / time. But since the momentum is 0, we can see that the average force exerted on the body is also 0. Looks like the knees did a great job at cushioning the landing!

d) Now, let's compare the stopping force to the performer's weight. The performer's weight is given by the equation weight = mass × acceleration due to gravity. Assuming we are on Earth, the acceleration due to gravity is approximately 9.8 m/s^2. Multiplying the weight by this value, we can find the force exerted by gravity. When we compare this force to the stopping force (which we already determined to be 0), we can see that the stopping force is much, much smaller. I guess gravity is the real superhero here!

To solve this problem, we'll use the principles of momentum, impulse, and force. We know the mass and height of the dancer, as well as the time it takes to stop. Let's go step-by-step to answer each part of the question:

a) With what momentum does the dancer reach the ground?

The momentum of an object is defined as the product of its mass and velocity. In this case, we want to find the momentum just before the dancer lands on the ground. Since the height is given, we can calculate the velocity of the dancer just before impact using the equation for gravitational potential energy:

Potential Energy = mgh

mgh = (1/2)mv^2

Where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), h is the height, and v is the velocity.

Rearranging the equation, we get:

v = sqrt(2gh)

Substituting the given values:

v = sqrt(2 * 9.8 m/s^2 * 0.32 m) ≈ 2.84 m/s

Now, we can find the momentum by multiplying the mass by the velocity:

Momentum = mass * velocity

Momentum = 65 kg * 2.84 m/s

b) What impulse is needed to make a stop?

Impulse is defined as the product of force and time. When the dancer stops, the impulse required to bring the dancer to rest is equal to the change in momentum:

Impulse = change in momentum

Since the dancer starts with a certain momentum before stopping and ends with zero momentum, the change in momentum is equal to the initial momentum:

Impulse = momentum

Impulse = 65 kg * 2.84 m/s

c) As the dancer lands, the knees bend, lengthening the stopping time to 0.050 s. Find the average force exerted on the body.

To find the average force exerted on the body, we need to divide the impulse by the time over which it acts:

Average Force = Impulse / Time

Average Force = (65 kg * 2.84 m/s) / 0.050 s

d) Compare the stopping force to the performer's weight.

To compare the stopping force to the performer's weight, we can divide the average force by the weight of the dancer:

Comparison = Average Force / Weight

Comparison = ((65 kg * 2.84 m/s) / 0.050 s) / 65 kg

Now, let's calculate the values we obtained in each step:

To solve these problems, we need to use the principles of momentum and impulse. Let's go through each part step by step:

a) Momentum is equal to the product of an object's mass and its velocity. In this case, we are given the mass of the dancer, which is 65 kg. However, we don't have the velocity directly, but we can calculate it using the kinematic equation for vertical motion. The equation we can use is:

v^2 = u^2 + 2gh

where:
- v is the final velocity (we can assume it to be 0 since the dancer reaches the ground),
- u is the initial velocity (also 0 since the dancer starts from rest),
- g is the acceleration due to gravity (approximately 9.8 m/s^2), and
- h is the height of the leap, which is given as 0.32 m.

Plugging in the values, the equation becomes:

0 = 0 + 2(9.8)(0.32)
0 = 6.272

Therefore, the velocity of the dancer just before reaching the ground is approximately 7.9 m/s. Now we can calculate the momentum:

momentum = mass × velocity
momentum = 65 kg × 7.9 m/s
momentum ≈ 514.5 kg·m/s

Thus, the momentum with which the dancer reaches the ground is approximately 514.5 kg·m/s.

b) Impulse is equal to the product of force and time. To bring the dancer to a stop, the impulse required will be equal in magnitude but opposite in direction to the initial momentum. So, we can use the equation:

impulse = change in momentum

Since the initial momentum is 514.5 kg·m/s and the final momentum is 0 kg·m/s, the change in momentum is -514.5 kg·m/s. Therefore:

impulse = -514.5 kg·m/s

The impulse needed to make the dancer stop is approximately -514.5 kg·m/s.

c) To find the average force exerted on the dancer, we can rearrange the equation for impulse: impulse = force × time. We are given the time, which is 0.050 s. Plugging in the values:

-514.5 kg·m/s = force × 0.050 s

Solving for force:

force = -514.5 kg·m/s ÷ 0.050 s
force = -10290 N

Thus, the average force exerted on the body is approximately -10290 N (negative sign indicates force in the opposite direction of the initial motion).

d) To compare the stopping force to the performer's weight, we can calculate the weight using the formula:

weight = mass × acceleration due to gravity
weight = 65 kg × 9.8 m/s^2
weight ≈ 637 N

Comparing this to the stopping force, we can see that the stopping force (-10290 N) is much greater than the performer's weight (637 N).

So, in summary:
a) The momentum with which the dancer reaches the ground is approximately 514.5 kg·m/s.
b) The impulse needed to make the dancer stop is approximately -514.5 kg·m/s.
c) The average force exerted on the body is approximately -10290 N.
d) The stopping force (-10290 N) is much greater than the performer's weight (637 N).