Projectile Motion Arrow Problem

In summary, the conversation discusses the calculation of the maximum height and distance of an arrow that was shot at an angle of 31.0° above the horizontal with a velocity of 53 m/s and hitting a target at the same height it was shot from. Neglecting air resistance, the acceleration in the x direction is 0 and the acceleration in the y direction is -9.8. The formula used to calculate the maximum height is Max Height = Vi*T+1/2*-9.8*T^2, with T being equal to 2.18 determined by [53*sin(40)]/9.8. The kinematic equation used to relate all the given quantities is unknown.
  • #1
davetheant
5
0
An arrow is shot at 31.0° above the horizontal. Its velocity is 53 m/s, and it hits the target.
A) What is the maximum height the arrow will attain?
B) The target is at the height from which the arrow was shot. How far away is it?


It says to neglect air resistance so acceleration in the x direction must be 0. Obviously gravity makes the acceleration in the y direction -9.8

I tried to calculate the maximum height by using the formula Max Height = Vi*T+1/2*-9.8*T^2

I got T to equal 2.18 by [53*sin(40)]/9.8
please help
 
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  • #2
In the first case, the initial velocity in the upward direction is known, acceleration is given and final velocity at the maximum height is known(?). Find the maximum height. Which kinematic equation relates all these quantities?
 
  • #3


I can confirm that your calculation for the maximum height is correct. To find the maximum height of the arrow, we can use the kinematic equation: h = h0 + V0y * t + 1/2 * a * t^2, where h0 is the initial height, V0y is the initial vertical velocity, a is the acceleration due to gravity, and t is the time.

In this case, h0 = 0 (since the arrow is shot from ground level), V0y = 53*sin(31) = 27.5 m/s, and a = -9.8 m/s^2. Plugging these values into the equation and solving for t, we get t = 2.18 seconds.

Substituting this value back into the equation, we get h = 27.5 * 2.18 + 0.5 * (-9.8) * (2.18)^2 = 75.4 meters. Therefore, the maximum height the arrow will attain is 75.4 meters.

To find the distance to the target, we can use the same kinematic equation for the horizontal motion: x = x0 + V0x * t, where x0 is the initial horizontal position, V0x is the initial horizontal velocity, and t is the time.

In this case, x0 = 0 (since the arrow is shot from ground level), V0x = 53*cos(31) = 45.6 m/s, and t = 2.18 seconds (as calculated earlier). Substituting these values into the equation, we get x = 45.6 * 2.18 = 99.3 meters. Therefore, the target is 99.3 meters away from the initial position of the arrow.

It is important to note that these calculations are based on the assumption of no air resistance, which may not be entirely accurate in real-life situations. Nonetheless, this is a good approximation for understanding the basic principles of projectile motion.
 

1. What is projectile motion and how does it relate to an arrow problem?

Projectile motion is the motion of an object projected into the air at an angle. It is a combination of horizontal motion (along the x-axis) and vertical motion (along the y-axis). In an arrow problem, we are considering the motion of an arrow that is shot into the air at an angle.

2. How is the initial velocity of the arrow determined?

The initial velocity of the arrow can be determined by measuring the speed of the arrow as it is launched and the angle at which it is launched. Using trigonometry, we can calculate the horizontal and vertical components of the initial velocity, which together make up the total initial velocity.

3. What factors can affect the trajectory of the arrow?

The trajectory of the arrow can be affected by a number of factors, including the initial velocity, the angle at which it is launched, air resistance, and gravity. Wind and other external forces can also impact the arrow's trajectory.

4. How can we calculate the maximum height and range of the arrow?

The maximum height and range of the arrow can be calculated using equations of motion, such as the vertical motion equation and the horizontal motion equation. These equations take into account the initial velocity, angle of launch, and acceleration due to gravity, among other factors.

5. Can we use projectile motion equations to accurately predict the motion of an arrow?

While projectile motion equations provide a good estimate of the motion of an arrow, they do not take into account all factors that can affect the arrow's trajectory. In real-world scenarios, air resistance and other external forces can impact the arrow's motion, making it difficult to predict with complete accuracy. However, these equations can provide a good approximation of the arrow's motion and are a useful tool for understanding and analyzing projectile motion problems.

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