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Will having only ONE new tire cause pull to the left on acceleration?

RSTP5200

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My '17 EB convertible is my daily driver, with around 45k miles on it, and I'm lovin it. I ran over something that trashed my rear tire, driver's side. I replaced the tire with a matching Goodyear Eagle F1 Asymmetric All Season, 'cause I've liked them. Goodyear folks said I had between 10k and 20k miles left on the tread on the remaining three. I drive it pretty hard, so 10 is probably closer to right.

Once I got on the highway with the new tire, I noticed that hard acceleration would cause a dramatic jerk to the left. Subsequently, steering to keep straight and then letting off the throttle causes a pull to the right. I initially noticed it the first time I really put my foot in it to pass someone going about 45. The car tracks fine under constant throttle, but accel/decel will get your attention. It's not a tire pressure issue. The issue seems less sever when accelerating from a standing start than from speeds around 40 mph and up.

Wondering if something may have been damaged, I had my local Ford dealership charge me $90 to "diagnose." They said nothing was damaged or bent, but that since the new tire was...well...new, with different circumference, etc., the handling issue was due to having one new tire and three old. Now the Goodyear folks are raising an eyebrow at this, as am I. While I know it's best to get tires in pairs (at least) or sets, I've never known running a brand new tire along with older ones to cause the kind of handling issue I'm seeing.

Ford says I need to buy 3 more new tires and have the alignment done, and that will fix the problem, along with my slush fund. I don't want to get rid of 3 tires prematurely. New ones are not cheap. Is somebody blowing smoke up my kilt here?

Thanks all.
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BmacIL

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Yes it can.
 

NightmareMoon

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At least consider another new rear tire. That (maybe plus an alignment) should solve the pull
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