Momentum, impulse and projectile motion – problems and solutions
1. A 0.2-kg ball will be inserted into hole C, as shown in the figure below. Hitter strikes the ball in 0.01 second and the path of B-C traveled in 1 second. Determine the magnitude of the force so the ball can be inserted into hole C. Acceleration due to gravity is 10 m/s2.
Angle (θ) = 60o
Mass of ball (m) = 0.2 kg
Acceleration due to gravity (g) = 10 m/s2
Time interval (Δt) = 0.01 second
Time interval to travel path B-C (t) = 1 second
Wanted : Force (F)
Equation of impulse : I = F Δt
Equation of the change in momentum : Δp = m (vt – vo).
I = Δp
F Δt = m (vt – vo)
F = m (vt – vo) / Δt
Δt = 0.01 second
m = 0.2 kg
vt = the final speed in the equation of impulse-momentum = the initial speed of ball (vo) in the projectile motion
vo = the initial speed in the equation of impulse-momentum = 0 m/s (initially ball at rest)
F = m (vt – vo) / Δt
F = 0.2 (vt – 0) / 0.01
F = 0.2 vt / 0,01
Determine the initial speed of the ball
Since the ball is hit until the ball reaches the point B = part 1 of the projectile motion.
The ball travels from point B to C = part 2 of the projectile motion.
Part 2 of the projectile motion :
The projectile motion could be understood by analyzing the horizontal and vertical component of the motion separately. The x motion occurs at a constant velocity and the y motion occurs at a constant acceleration of gravity.
Horizontal distance (x) = 5 meters
Time in air (t) = 1 second
x and t are known so that vox can calculated using the equation of the uniform linear motion
vox = x / t = 5 meters / 1 second = 5 m/s.
Part 1 of the projectile motion :
The horizontal component of speed, vox is always same so vox in part 1 of the projectile motion = vox in part 2 of the projectile motion = 5 m/s.
vox = 5 m/s
θ = 60o
vox and θ are known so the initial speed (vo) can be calculated.
cos θ = adj / hyp
cos θ = vox / vo
vo = vox / cos θ = 5 / cos 60o = 5 / 0.5 = 10 m/s
The initial speed (vo) is 10 m/s.
The initial speed of the ball (vo) in projectile motion = the final speed of the ball (vt) in the equation of impulse-momentum.
Determine the magnitude of force (F)
F = 0.2 vt / 0.01
F = 0.2 (10) / 0.01
F = 2 / 0.01
F = 200 Newton