Ballistic Launcher

A steel ball is fired from a ballistic launcher at different angles. The launched ball has been found to travel from the edge of a table to land 22.1 \(cm\) from the far end of the table when starting from the height of the table and launched at an angle of \(^{\circ}\) above the horizontal. When launched at \(^{\circ}\), the ball easily clears the table to land on the floor.

You may assume that \(t=0\) at the instant that the ball leaves the launcher.

Part 1

Ignoring air resistance, write an expression for the \(x\)-component of the acceleration of the ball while it is in the air.

Answer Section

Please enter in a numeric value in \(m/s^2\).

Part 2

Ignoring air resistance, write an expression for the \(y\)-component of the acceleration of the ball while it is in the air.

Answer Section

Please enter in a numeric value in \(m/s^2\).

Part 3

Write an expression for the full acceleration vector in terms of the unit vectors \(\hat{x}\) and \(\hat{y}\).

You should provide a symbolic answer in terms of the following variables: \(\hat{x}\), \(\hat{y}\), and \(g\).

Note that it may not be necessary to use every variable. Use the following table as a reference for using each variable.

\(Variable\)	Use
\(g\)	g
\(\hat{x}\)	x_hat
\(\hat{y}\)	y_hat

Answer Section

Please enter in a symbolic answer.

Part 4

On its maximum setting, the speed of the ejected steel ball is \(v\_{max}\).

Write an expression for the \(x\)-component of the velocity of the ball as a function of time when the angle of launch is \(\theta^{\circ}\).

You should provide a symbolic answer in terms of the following variables: \(t\), \(v\_{max}\), \(\theta\), \(\hat{x}\), \(\hat{y}\), and \(g\).

Note that it may not be necessary to use every variable. Use the following table as a reference for using each variable.

\(Variable\)	Use
\(t\)	t
\(v\_{max}\)	v_max
\(\theta\)	theta
\(g\)	g

Answer Section

Please enter in a symbolic answer.

Part 5

On its maximum setting, the speed of the ejected steel ball is \(v\_{max}\).

Write an expression for the \(y\)-component of the velocity of the ball as a function of time when the angle of launch is \(\theta^{\circ}\).

You should provide a symbolic answer in terms of the following variables: \(t\), \(v\_{max}\), \(\theta\), \(\hat{x}\), \(\hat{y}\), and \(g\).

Note that it may not be necessary to use every variable. Use the following table as a reference for using each variable.

\(Variable\)	Use
\(t\)	t
\(v\_{max}\)	v_max
\(\theta\)	theta
\(g\)	g

Answer Section

Please enter in a symbolic answer.

Part 6

Write an expression for the full velocity vector as a function of time.

You should provide a symbolic answer in terms of the following variables: \(v_x\) and \(v_y\).

Note that it may not be necessary to use every variable. Use the following table as a reference for using each variable.

\(Variable\)	Use
\(v\_{x}\)	v_x
\(v\_{y}\)	v_y
\(\hat{x}\)	x_hat
\(\hat{y}\)	y_hat

Answer Section

Please enter in a symbolic answer.

Part 7

On its maximum setting, the speed of the ejected steel ball is \(v\_{max} = \) 4.36 \(m/s\).

From your expression for \(v\_{y,\theta^{\circ}}\), solve for the time that the ball was in the air, when the ball was launched at a \(^{\circ}\) angle.

Hint: what would be the vertical component of the velocity of the ball just before it hit the table when it returns to the same height from which it left?

Answer Section

Please enter in a numeric value in \(s\).

Part 8

Letting the edge of the table from which the ball was launched have coordinates \((x_0, y_0) = (0, 0)\), write an expression for the \(x\)-component of the object’s position vector.

You should provide a symbolic answer in terms of the following variables: \(t\), \(x_0\), \(y_0\), \(v\_{0_x}\), \(v\_{0_y}\), and \(g\).

Note that it may not be necessary to use every variable. If a value is zero, do not include it. Use the following table as a reference for using each variable.

\(Variable\)	Use
\(t\)	t
\(x_0\)	x0
\(y_0\)	y0
\(v\_{0_x}\)	v_x
\(v\_{0_y}\)	v_y
\(g\)	g

Answer Section

Please enter in a symbolic answer.

Part 9

Letting the edge of the table from which the ball was launched have coordinates \((x_0, y_0) = (0, 0)\), write an expression for the \(y\)-component of the object’s position vector.

You should provide a symbolic answer in terms of the following variables: \(t\), \(x_0\), \(y_0\), \(v\_{0_x}\), \(v\_{0_y}\), and \(g\).

Note that it may not be necessary to use every variable. If a value is zero, do not include it. Use the following table as a reference for using each variable.

\(Variable\)	Use
\(t\)	t
\(x_0\)	x0
\(y_0\)	y0
\(v\_{0_x}\)	v_x
\(v\_{0_y}\)	v_y
\(g\)	g

Answer Section

Please enter in a symbolic answer.

Part 10

Using your results from Part 8 and/or Part 9, estimate the total length of the table. Note that the steel ball landed about 22.1 \(cm\) from the end of the table when fired at a \(^{\circ}\) angle.

Answer Section

Please enter in a numeric value in \(m\).

Attribution

Problem is licensed under the CC-BY-NC-SA 4.0 license.