Effect of speed on horizontal displacement in projectile motion

In summary: If that angle is θ, then the horizontal component of speed is vcosθ. In this problem, we are told that the two trajectories have the same height. That means that the two trajectories have the same time of flight. This means that the vertical component of speed must be the same for both trajectories. But the vertical component of speed for the second trajectory is only half the value of the first trajectory. This means that the angle is different. Since the angle is different, the horizontal component of speed is different. It is not half of its original value. In summary, when throwing a ball at half the original speed, the range will not be halved as the horizontal component of speed is
  • #1
darla1608
2
0

Homework Statement



Question: Provided all else is equal, if a ball is thrown at half the original speed will it travel half as far?


Homework Equations



Δx = vΔt


The Attempt at a Solution



In projectile motion, the time of flight will be the same no matter the horizontal velocity, as long as the height is the same. When I plug two different velocities into the above equation ( one being half of the other), the displacement is half.

Issue with this? The answer in the book states this is not true. What am I doing wrong or not considering?
 
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  • #2
darla1608 said:

Homework Statement



Question: Provided all else is equal, if a ball is thrown at half the original speed will it travel half as far?

Homework Equations



Δx = vΔt

The Attempt at a Solution



In projectile motion, the time of flight will be the same no matter the horizontal velocity, as long as the height is the same. When I plug two different velocities into the above equation ( one being half of the other), the displacement is half.

Issue with this? The answer in the book states this is not true. What am I doing wrong or not considering?

Range is proportional to the square of the launch speed and to the sine of twice the launch angle.

If you keep the same angle but halve the speed, the range will be decrease by a factor of 1/4.

What you appear to be trying to do is launch the projectile with the same vertical speed (so it will reach the same height) but less horizontal speed. This means it will have lower launch energy and a larger launch angle (relative to the horizontal).

AM
 
Last edited:
  • #3
Let Δt be the time it takes for the projectile to travel.

The equation Δx = vΔt only applies for one-dimensional kinematics. In 2D kinematics, the v is the x component of velocity. Just because velocity has been halved doesn't mean that the x component of velocity has been halved.
 
  • #4
thank you
 
  • #5
"Speed" is not the same as "horizontal component of speed". To go from one to the other you need to know the angle the trajectory makes with the horizontal.
 

Related to Effect of speed on horizontal displacement in projectile motion

1. What is projectile motion?

Projectile motion is the motion of an object through the air that is affected by gravity. It follows a curved path called a parabola. An example of projectile motion is a ball being thrown into the air.

2. How does speed affect horizontal displacement in projectile motion?

As the speed of the object increases, the horizontal displacement also increases. This is because the object is able to travel further in the same amount of time due to its higher velocity. However, the angle of projection also affects the horizontal displacement.

3. What is the relationship between speed and horizontal displacement in projectile motion?

The relationship between speed and horizontal displacement in projectile motion is directly proportional. This means that as the speed of the object increases, the horizontal displacement also increases, assuming all other factors remain constant.

4. How can the effect of speed on horizontal displacement be calculated?

The effect of speed on horizontal displacement can be calculated using the equation D = v * t, where D is the horizontal displacement, v is the speed, and t is the time. This equation assumes a constant speed and does not take into account factors such as air resistance.

5. How does air resistance affect the effect of speed on horizontal displacement in projectile motion?

Air resistance can decrease the effect of speed on horizontal displacement in projectile motion. This is because air resistance acts as a force that opposes the motion of the object, causing it to slow down and travel a shorter distance. In real-life scenarios, air resistance must be taken into account when calculating the effect of speed on horizontal displacement.

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