Effect of speed on horizontal displacement in projectile motion

[darla1608]

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?

 

[Andrew Mason]

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

 

[Yenaled]

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.

 

[darla1608]

thank you

 

[HallsofIvy]

"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.