Question

A dart gun is fired while being held horizontally at a height of 1.00 m above ground level and while it is at rest relative to the ground. The dart from the gun travels a horizontal distance of 5.00 m. A college student holds the same gun in a horizontal position while sliding down a 45.0° incline at a constant speed of 2.00 m/s. How far will the dart travel if the student fires the gun when it is 1.00 m above the ground?

Answer 1

IDK

Answer 2

Sketch:

Below sketch shows the free body diagram of a dart gun fired horizontally from a height 1 m above the ground and travels a horizontal distance of 5.0 m.

Solution:

Dart gun:

If is the time taken, and v is the velocity of the dart to reach the ground, then equation for time in terms of height can be written using the kinematic equation,

For the given problem,

[Initial velocity of the dart gun is zero along the vertical direction, vertical distance, and acceleration of the dart gun ]

To obtain time taken reach the ground, substitute 1 m for, for in the equation

To find the velocity of projection , use the below equation:

Here, horizontal distance travelled by dart gun.

College student:

Initial speed of the student,

As student moves along the inclined plane, the vertical distance travelled by him is given by a relation,

Here, is the acceleration of the studentand is the angle of inclination, time of travel.

Substitute for , 1 m for, and for in the equation

The horizontal distance traveled by the dart in the second case,

The dart travel a final distance if the student fire the gun when it is 1 m above the ground

Therefore, dart travel a final distance if the student fire the gun when it is 1 m above the ground is 4.2 m.

Answer 3

Use the following kinematical equations to find the distance travelled by the dart.

The equation which gives the relation between distance travelled S and time t, when the object is projected with an initial velocity v is,

Here, a is the acceleration.

The equation which relates the initial velocity, final velocity and the distance travelled is,

Here, a is the acceleration.

The vertical distance travelled by the dart is given by,

In the vertical motion of the dart, the acceleration of the dart is due to the acceleration due to gravity.

The dart is fired horizontally from a gun which is held stationary. Hence the initial vertical velocity is zero. Thus, the time of flight is given by,

Substitute for y and for g.

The initial speed of the dart relative to the gun is,

Here, x is the horizontal distance travelled by the dart and t is the time.

Substitute for x and for t.

The gun moves down the incline with speed . Thus, the resultant velocity of the dart is as shown in the following vector diagram. The angle of inclination is .

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The vertical component of the velocity of the dart is,

Here, is the angle of inclination.

Substitute for and for.

The horizontal component of the dart is,

Here, is the speed of the dart relative to the gun.

Substitute for, for, and for.

The final vertical velocity of the dart after it is dropped from a height y is,

Substitute for, for g and for.

Therefore, the time of flight is,

Substitute for, for, and for g.

The horizontal distance travelled by the dart is,

Substitute for and for t.

Therefore, the distance travelled by the dart when it is fired by the student is.