Calculating Horizontal Distance of Dart Fired by Child on Incline

In summary, the problem involves a dart gun being fired horizontally from a height of 1.55 m above ground level and traveling a horizontal distance of 7.77 m. A child is then sliding down a 39.7 degree incline at a constant speed of 0.869 m/s while holding the same gun in a horizontal position. The question is asking for the horizontal distance the dart will travel when the child fires the gun at a height of 1.1 m above the ground. To solve this, the child's x-direction velocity can be found by using sin 39.7 * 0.869 m/s. The muzzle velocity of the dart gun can also be determined using the information provided. It is not
  • #1
ngoldfarb
1
0
Okay here is the problem:
A dart gun is fired while being held horizontally at a height of 1.55 m above ground level, and at rest relative to the ground. The dart from the gun travels a horizontal distance of 7.77 m. A child holds the same gun in a horizontal position while sliding down a 39.7 degree incline at a constant speed of 0.869 m/s. The acceleration of gravity is 9.8 m/s^2.

What horizontal distance "x" will the dart travel if the child fires the gun forward when it is 1.1m above the ground? Answer in units of m.

I tried finding out how fast the boy would be traveling in the x direction by using: sin 39.7 * 0.869 m/s but once I have that information, I don't know what to do with it. thanks.
 
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  • #2
ngoldfarb said:
I tried finding out how fast the boy would be traveling in the x direction by using: sin 39.7 * 0.869 m/s but once I have that information, I don't know what to do with it. thanks.
Should you use the sine or cosine to find the x-component of the child's velocity?

From the information provided in the first part, can you determine the muzzle velocity of the dart gun?
 
  • #3


I would approach this problem by first identifying the relevant variables and equations that can be used to solve for the horizontal distance traveled by the dart. In this case, the variables are the initial height of the dart (1.55 m), the initial velocity of the dart (0 m/s), the acceleration of gravity (9.8 m/s^2), the angle of the incline (39.7 degrees), and the constant speed of the child (0.869 m/s). The equation that relates these variables is the projectile motion equation, which is: x = v0*cos(theta)*t, where x is the horizontal distance, v0 is the initial velocity, theta is the angle of the incline, and t is the time of flight.

To solve for the horizontal distance, we need to find the time of flight. This can be done by using the equation for the vertical distance traveled by the dart, which is: y = v0*sin(theta)*t - 1/2*g*t^2, where y is the vertical distance and g is the acceleration of gravity. Since we know that the dart will land at a height of 1.1 m, we can set y equal to 1.1 m and solve for t.

1.1 m = 0 + 0.869 m/s * sin(39.7 degrees) * t - 1/2 * 9.8 m/s^2 * t^2

Solving for t, we get t = 0.388 seconds. Now, we can plug this value into the horizontal distance equation to find the distance traveled by the dart:

x = 0 + 0.869 m/s * cos(39.7 degrees) * 0.388 seconds = 0.592 m

Therefore, the dart will travel a horizontal distance of approximately 0.592 m when fired by the child from a height of 1.1 m on the incline. It is important to note that this is an ideal calculation and does not take into account any external factors such as air resistance or friction on the incline.
 

1. What is the formula for calculating horizontal distance of a dart fired by a child on an incline?

The formula for calculating horizontal distance is: d = v*cos(theta)*t, where d is the horizontal distance, v is the initial velocity of the dart, theta is the angle of the incline, and t is the time the dart is in flight.

2. How do you determine the initial velocity of the dart?

The initial velocity of the dart can be determined by measuring the distance the dart travels horizontally and the time it takes to travel that distance. Then, using the formula v = d/t, the initial velocity can be calculated.

3. Is the angle of the incline important in calculating the horizontal distance?

Yes, the angle of the incline is a crucial factor in calculating the horizontal distance. As the angle increases, the horizontal distance also increases, and vice versa.

4. Can the horizontal distance be calculated accurately in real-life situations?

The calculation of horizontal distance is a simplified model and may not be accurate in real-life situations due to factors such as air resistance and varying initial velocities. However, it can give a rough estimate of the distance traveled by the dart.

5. How can this calculation be useful in real-life applications?

This calculation can be useful in various fields such as sports, engineering, and physics. It can help engineers determine the trajectory of projectiles and assist athletes in improving their throwing or shooting skills.

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