A student performs a ballistic pendulum experiment, using an...

A student performs a ballistic pendulum experiment, using an apparatus similar to that shown in Figure 9.11b. She obtains the following average data: $h=$ $8.68 \mathrm{~cm}, m_{1}=68.8 \mathrm{~g}$, and $m_{2}=263 \mathrm{~g} .$ The symbols re- fer to the quantities in Figure 9.11a. (a) Determine the initial speed $v_{1 i}$ of the projectile. (b) In the second part of her experiment she is to obtain $v_{1 i}$ by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its horizontal displacement $x$ and vertical displacement $y$ (Fig. P9.70). Show that the initial speed of the projectile is related to $x$ and y through the relationship $$ v_{1 i}=\frac{x}{\sqrt{2 y / g}} $$ What numerical value does she obtain for $v_{1 i}$ on the basis of her measured values of $x=257 \mathrm{~cm}$ and $y=$ $85.3 \mathrm{~cm} ?$ What factors might account for the difference in this value compared with that obtained in part (a)?

A student performs a ballistic pendulum experiment, using an apparatus similar to that shown in Figure 9.11b. She obtains the following average data: $h=$ $8.68 \mathrm{~cm}, m_{1}=68.8 \mathrm{~g}$, and $m_{2}=263 \mathrm{~g} .$ The symbols re-
fer to the quantities in Figure 9.11a. (a) Determine the initial speed $v_{1 i}$ of the projectile. (b) In the second part of her experiment she is to obtain $v_{1 i}$ by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its horizontal displacement $x$ and vertical displacement $y$ (Fig. P9.70). Show that the initial speed of the projectile is related to $x$ and y through the relationship
$$
v_{1 i}=\frac{x}{\sqrt{2 y / g}}
$$
What numerical value does she obtain for $v_{1 i}$ on the basis of her measured values of $x=257 \mathrm{~cm}$ and $y=$ $85.3 \mathrm{~cm} ?$ What factors might account for the difference in this value compared with that obtained in part (a)?

Key Concepts

Ballistic Pendulum Experiment

The ballistic pendulum is a classic experimental setup that combines collision dynamics with pendular motion to measure high velocities. This apparatus allows one to determine a projectile's speed by analyzing the energy transformation that occurs when the projectile embeds itself in a pendulum and causes it to swing upward.

Conservation of Momentum

In the context of the ballistic pendulum, conservation of momentum is applied during the inelastic collision when the projectile strikes and sticks to the pendulum bob. Since no external horizontal forces are acting during the collision, the total momentum before and after the collision remains constant, making it a fundamental principle for determining the initial speed of the projectile.

Conservation of Energy

After the collision, the kinetic energy of the combined system is transformed into gravitational potential energy as the pendulum rises. The conservation of energy principle connects the measurable height gained by the pendulum to the kinetic energy immediately after the collision, thereby providing a way to back-calculate the projectile’s initial speed.

Projectile Motion Kinematics

The study of projectile motion involves analyzing objects launched into the air and moving under the influence of gravity. Kinematic equations relate horizontal displacement, vertical displacement, and time of flight, which are essential for deriving relationships between these parameters and the initial velocity when the projectile is fired horizontally.

Experimental Uncertainty

Experimental uncertainty refers to the differences in measurements that arise from errors inherent in practical setups. Factors such as air resistance, friction, alignment imperfections, and measurement inaccuracies can lead to discrepancies between values obtained through different methods, highlighting the importance of considering error analysis in experimental physics.

Transcript

00:01 In this one, we're going to be able to relate the results from the ballistic pendulum to the results that were to find the initial velocity of a bullet that was fired at a block that's hanging from a pendulum that is on a pendulum.

00:18 And the bullet will embed itself in the block and will swing up to some height h.

00:25 And we know from the previous example how to relate the initial velocity of the bullet to that height.

00:32 And we're going to get a result based on some data that was given, and we're going to relate that to the same experiment done.

00:41 But instead of using the ballistic pendulum, we're just going to use projectile motion to calculate the initial velocity of the bullet, and we're going to compare the answers.

00:49 So first, we're going to plug in the values given into the previously known ballistic pendulum equation.

00:57 And so we need to plug in the mass of each object for which we need to convert to kilograms.

01:05 So we divide by a thousand in order to get from grams to kilograms.

01:10 So the mass of the bullet will be 0 .0688 kilograms plus the mass of the block, which is 0 .263 kilograms, and divided by 0 .0688 kilograms.

01:25 All that times the square root of 2 times g, which is 9 .81 .2.

01:28 Meters per second squared times h which we need to convert to meters to get from centimeters to meters you divide by a hundred so this is 0 .0 868 meters you plug this into a calculator you'll find that it's approximately 6 .163 meters per second apologies 6 .3 right ahead of myself approximately 6 .3 meters per second that's our first result using the ballistic pendulum.

02:04 If we use projectile motion equations, we know that from kinematic equations and projectile motion, that the equation of motion for the y -coordinate of anything moving through the air will be negative one -half g, g squared, plus a term that relates to the initial velocity in the y direction.

02:28 But in this case, our bullet is being fired exactly horizontal, so there is no initial velocity in the y direction.

02:37 So that second term would be zero, so there's zero, plus the initial height.

02:44 We do the same thing with the x.

02:46 We know there's no acceleration in the x direction.

02:49 We're neglecting air resistance.

02:51 And so this this will be the initial velocity in the x direction, which we can just label the same thing that we labeled the answer over here, v.

03:03 Sub 1i times t, plus the initial x position.

03:09 And we can move our coordinate system such that this is zero.

03:14 We can start at the origin, wherever the x coordinate of the bullet, wherever it starts, can be zero.

03:21 And so we want to know what this is.

03:25 And what we can measure is the range of the projectile, which we also labeled as x.

03:33 And we can also measure the initial height, which i will call y sub zero, and i'll call this max for the range...