Calculate Min Initial Kinetic Energies for Javelin, Discus, Shot Throws

[alevis]

Homework Statement

The masses of the javelin, discus, and shot are 1.0 kg, 2.0 kg, and 8.0 kg, respectively, and record throws in the

corresponding track events are about 98 m, 75 m, and 25 m, respectively. Neglecting air resistance,
(a) calculate the minimum initial kinetic energies that would produce these throws, and
(b) estimate the average force exerted on each object during the throw, assuming the force acts over a distance

of 2.0 m.
(c) Do your results suggest that air resistance is an important factor?

Homework Equations

KE = 1/2mv²
PE = mgh

The Attempt at a Solution

1/2mv² = mgh.
stuck!

 

[chrisk]

Consider the path of the objects. Each is a parabola, and there are two different launch angles (theta +/- 45 degrees) that can produce the same horizontal distance of travel with the exception of a 45 degree launch angle. A particular launch angle will require the minimum amount of kinetic energy (initial velocity). Determine this particular launch angle, then the velocity required to produce the horizontal distance for each object can be determined. Use the frictionless kinematic equations for a projectile to find the velocities.

 

[thomate1]

I would approach this problem by first identifying the relevant equations and variables. In this case, we are dealing with kinetic energy (KE) and potential energy (PE). The mass (m) and velocity (v) of each object will be important in calculating their kinetic energy, while the height (h) and acceleration due to gravity (g) will be important in calculating their potential energy.

To start, let's calculate the minimum initial kinetic energy for each object that would produce the given throws. We can use the equation KE = 1/2mv^2, where m is the mass of the object and v is the velocity. For the javelin, discus, and shot, this would be:

KEjavelin = 1/2 * 1.0 kg * (98 m/s)^2 = 4801 J
KEdiscus = 1/2 * 2.0 kg * (75 m/s)^2 = 5625 J
KEshot = 1/2 * 8.0 kg * (25 m/s)^2 = 2000 J

Next, we can estimate the average force exerted on each object during the throw. The equation for average force is F = Δp/Δt, where Δp is the change in momentum and Δt is the time it takes for that change to occur. In this case, we can assume that the force acts over a distance of 2.0 m, so we can use the equation F = mΔv/Δt, where m is the mass of the object and Δv is the change in velocity.

For the javelin, discus, and shot, this would be:
Fjavelin = (1.0 kg * 98 m/s) / 2.0 m = 49 N
Fdiscus = (2.0 kg * 75 m/s) / 2.0 m = 75 N
Fshot = (8.0 kg * 25 m/s) / 2.0 m = 100 N

Finally, we can consider whether air resistance is an important factor in these throws. Neglecting air resistance means that we are assuming the objects are moving through a vacuum, which is not the case in the real world. In reality, air resistance would slow down the objects and decrease their kinetic energy. However, based on our calculations,