A projectile of mass 0.817 kg is shot straight up with an initial speed of 25.5 m/s. (a) How high would it go if there were no air resistance? (b) If the projectile rises to a maximum height of only 9.14 m, determine the magnitude of the average force due to air resistance.
a. V^2 = Vo^2 + 2g*h
h = (V^2-Vo^2)/2g
h = (0-(25.5))/-19.6 = 33.2 m.
33.2
To solve these problems, we will use the principles of projectile motion and the equations of motion. Let's break down each part of the problem step by step:
(a) How high would it go if there were no air resistance?
To find the maximum height reached by the projectile without air resistance, we can use the kinematic equation for vertical motion:
v_f^2 = v_i^2 + 2aΔy
Where:
v_f = final velocity (which is 0 at the highest point)
v_i = initial velocity (given as 25.5 m/s)
a = acceleration (due to gravity, which is -9.8 m/s^2)
Δy = change in height
Using the equation and solving for Δy, we get:
0 = (25.5 m/s)^2 + 2(-9.8 m/s^2) Δy
Δy = (25.5 m/s)^2 / (2 * 9.8 m/s^2)
Δy = 32.98 m
Therefore, without considering air resistance, the projectile would reach a maximum height of 32.98 meters.
(b) If the projectile rises to a maximum height of only 9.14 m, determine the magnitude of the average force due to air resistance.
To determine the magnitude of the average force due to air resistance, we need to find the total work done by the force of air resistance.
The work done by air resistance is equal to the change in mechanical energy of the particle. In this case, the change in mechanical energy is equal to the work done by gravity, which is equal to the change in potential energy.
The change in potential energy can be calculated using the formula:
ΔPE = m * g * Δy
Where:
m = mass of the projectile (given as 0.817 kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Δy = change in height (given as 9.14 m)
ΔPE = (0.817 kg) * (9.8 m/s^2) * (9.14 m)
ΔPE = 72.07 J
So, the change in potential energy of the projectile is 72.07 Joules.
Now, for a non-conservative force like air resistance, the work is equal to the change in kinetic energy. Therefore:
Work = ΔKE
Since the initial and final velocities are the same (0 m/s at the highest point), the change in kinetic energy is zero.
So, Work = ΔKE = 0
Therefore, the magnitude of the average force due to air resistance is zero.