The force on a .4 kg cannon ball in the barrel when fired at t=0 seconds is given by F= 1000- 5 * 10^4t. The acceleration of the ball is 0 as it leaves the barrel
The total impulse in Newton seconds on the ball during the .02 seconds it's in the barrel would be
To find the total impulse on the cannonball, we need to integrate the force function over the time interval it is in the barrel.
Given:
Mass of the cannonball (m): 0.4 kg
Force on the cannonball (F): 1000 - 5 * 10^4t
Acceleration of the cannonball (a): 0
Time in the barrel (t): 0.02 seconds
Impulse (J) is defined as the change in momentum. It is equal to the force applied to an object multiplied by the time over which it is applied. Mathematically, impulse can be calculated using the following formula:
J = ∫F dt
To find the total impulse, we integrate the force function F with respect to time from 0 to 0.02 seconds:
J = ∫(1000 - 5 * 10^4t) dt
Now, let's solve the integral:
J = ∫1000 dt - ∫(5 * 10^4t) dt
∫1000 dt = 1000t + C1 (where C1 is the constant of integration)
∫(5 * 10^4t) dt = (5 * 10^4 / 2) t^2 + C2 (where C2 is the constant of integration)
Now, substitute the limits of integration (0 and 0.02) and subtract:
J = (1000 * 0.02 + C1) - (5 * 10^4 / 2 * (0.02)^2 + C2)
J = (20 + C1) - (5 * 10^4 / 2 * 0.0004 + C2)
J = 20 - 5 * 10^4 * 0.0002 + (C1 - C2)
Since the acceleration of the cannonball is zero as it leaves the barrel, the impulse on the cannonball would be equal to its muzzle velocity times its mass. Therefore, the total impulse on the cannonball during the 0.02 seconds it's in the barrel would be 20 Newton-seconds.
To find the total impulse on the ball during the 0.02 seconds it's in the barrel, we need to integrate the force function over that time interval.
The impulse is the change in momentum, which is equal to the integral of force with respect to time. Mathematically, we have:
Impulse = ∫(F dt)
Given that the force function is F = 1000 - 5 * 10^4t, we need to integrate this equation with respect to time over the interval t = 0 to t = 0.02 seconds.
Impulse = ∫(F dt) [from t = 0 to t = 0.02]
Impulse = ∫((1000 - 5 * 10^4t) dt) [from t = 0 to t = 0.02]
To calculate the integral, we need to split it into two parts:
Impulse = ∫(1000 dt) - ∫(5 * 10^4t dt) [from t = 0 to t = 0.02]
Now we can integrate each part:
Impulse = 1000t - (5 * 10^4 * 0.5t^2) [from t = 0 to t = 0.02]
Evaluating this expression for t = 0.02 and t = 0 gives us:
Impulse = 1000(0.02) - (5 * 10^4 * 0.5(0.02)^2) - [1000(0) - (5 * 10^4 * 0.5(0)^2)]
Impulse = 20 - 10 - 0
Therefore, the total impulse on the ball during the 0.02 seconds it's in the barrel is 10 Newton-seconds.