Integrated Concepts

A toy gun uses a spring with a force constant of 300 N/m to propel a 10.0-g steel ball. If the spring is compressed 7.00 cm and friction is negligible: (a) How much force is needed to compress the spring? (b) To what maximum height can the ball be shot? (c) At what angles above the horizontal may a child aim to hit a target 3.00 m away at the same height as the gun? (d) What is the gun’s maximum range on level ground?

Part (a)

Step 1 of 9:

A steel ball is fired using a toy gun. The spring inside the gun is given and the force constant is also provided. Assume that the friction is negligible. We are going to find the force required to compress the spring.

The force constant of the spring

The displacement of the spring or

Step 2 of 9:

The force required to compress the spring

Substituting the values

The force required to compress the ball is -21.0 N

Part (b)

Step 3 of 9:

If the gun is shot vertically, the ball goes in the upward direction. Our aim is to find the maximum height that the ball can go. The expression for the vertical height of a projectile is used to find the maximum height here. Before finding the vertical height, we need to calculate the initial speed of the ball.

Step 4 of 9:

From conservation of energy

The potential energy of the ball by compressed spring = the kinetic energy of the ball

Where is the mass of the ball. or

is the initial speed of the ball when it is released

Solving the equation (2) for

Substituting the values

The initial speed of the ball is

Step 5 of 9:

The expression for the maximum height of a projectile (ball)

Where g is acceleration due to gravity and

, since the ball is shot vertically upward.

Therefore the maximum height is

or

The maximum height can be reached by the ball is 7.5 m

Part (c)