# 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 \(v_y\).

\(Variable\) |
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 objectâ€™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 objectâ€™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.