A student performs a ballistic pendulum experiment using an apparatus similar to

Question

A student performs a ballistic pendulum experiment using an apparatus similar to that shown in the first figure. She obtains the following average data: h = 8.38 cm, projectile mass m1 = 69.3 g, and pendulum mass m2 = 263 g. (a) Determine the initial speed v1A of the projectile. m/s (b) The second part of her experiment is to obtain v1A by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its horizontal position x and distance of fall y, as shown in the second figure. What numerical value does she obtain for v1A on the basis of her measured values of x = 255 cm and y = 87.4 cm? m/s (c) What factors might account for the difference in this value compared to that obtained in part (a)?

Explanation / Answer

For (a), assume no friction losses due to air resistance. Assume momentum is conserved.

Momentum before: m1vi + m2(0)

Momentum after: (m1 + m2)vf

m1v1 = (m1 + m2)vf

vi = (m1 + m2)vf/m1

Kinetic energy of the projectile and pendulum becomes potential energy as the combined mass rises the distance h.

(1/2)(m1 + m2)vf2 = (m1 + m2)gh

vf = (2gh) = (2(9.8 m/s2)(.0838 m)) = 1.28 m/s

vi = (.0693 kg + .263 kg)(1.28 m/s)/(.0693 kg) = 6.14 m/s

For part (b), use the displacement equation for by horizontal and vertical motion.

s = (1/2)at2 + v0t + s0

The origin is the launch point.

For the horizontal direction, a = 0, s0 = 0, v0 = vi

For the vertical direction, a = - g, s0 = y, v0 = 0

Solve the vertical displacement for the time when s = 0. At this same time, the horizontal displacement = x and initial velocity can be calculated.

Solving vertical displacement

y = - (1/2)gt2 + .0874

When y = 0 (projectile is on floor)

t = (2(.0874)/9.8) = .13 s

Solving horizontal displacement

x = vit

When t = 0.13, projectile is on floor .255 m from the table

.255 = vi(.13)

vi = .255/.13 = 1.96 m/s

For part (c), the projectile is constrained to travel in a circular arc, with a constantly changing velocity. We ignore angular acceleration. We ignore the direction of the linear velocity vector when the pendulum has risen a distance h. Although the horizontal component of linear velocity at h is less than the initial velocity when the projectile joined the pendulum, we calculated initial velocity as if it were the same as the linear velocity a distance h above the equilibrium position.

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